Difference between revisions of "Andrew Walker Problems"
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Show that <math>\{1+i,1-i \}</math> form a linearly independent set of vectors in <math>\mathbb{C}</math>, viewed as a vector space over <math>\mathbb{R}</math>. | Show that <math>\{1+i,1-i \}</math> form a linearly independent set of vectors in <math>\mathbb{C}</math>, viewed as a vector space over <math>\mathbb{R}</math>. | ||
| − | + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | |
| − | Recall that the set of vectors <math>\{v_{1},\ldots, v_{n} \}</math> in a vector space <math>V</math> (over a field <math>\mathbb{F}</math>) are said to be '''linearly independent''' if whenever <math>c_{1},\ldots,c_{n}</math> are scalars in <math>\mathbb{F}</math> such that <math>c_{1}v_{1} + \cdots + c_{n}v_{n} = 0,</math> then <math>c_{1} = \cdots = c_{n} = 0</math>. So for this problem, since we’re considering the complex numbers <math>\mathbb{C}</math> as a vector space over <math>\mathbb{R}</math>, we must show that whenever <math>c_{1},c_{2} \in \mathbb{R}</math> and <math>c_{1}(1+i) + c_{2}(1-i) = 0,</math> then <math>c_{1} = c_{2} = 0</math>. Rearranging the above equation, we obtain <math>(c_{1} + c_{2}) + (c_{1} - c_{2}) i = 0.</math> Now, a complex number is equal to <math>0</math> if and only if its real and imaginary parts are both <math>0</math>. So in this case, we conclude that <math>c_{1} + c_{2} = 0 \text{ and } c_{1} - c_{2} = 0.</math> This implies <math>c_{1} = c_{2}</math>, so that <math>c_{1} + c_{2} = 2c_{2} = 0</math>, which yields <math>c_{1} = c_{2} = 0</math>. Thus we conclude the vectors <math>1+i,1-i</math> are linearly independent in <math>\mathbb{C}</math> (over <math>\mathbb{R}</math>). | + | !Proof: |
| + | |- | ||
| + | |Recall that the set of vectors <math>\{v_{1},\ldots, v_{n} \}</math> in a vector space <math>V</math> (over a field <math>\mathbb{F}</math>) are said to be '''linearly independent''' if whenever <math>c_{1},\ldots,c_{n}</math> are scalars in <math>\mathbb{F}</math> such that <math>c_{1}v_{1} + \cdots + c_{n}v_{n} = 0,</math> then <math>c_{1} = \cdots = c_{n} = 0</math>. So for this problem, since we’re considering the complex numbers <math>\mathbb{C}</math> as a vector space over <math>\mathbb{R}</math>, we must show that whenever <math>c_{1},c_{2} \in \mathbb{R}</math> and <math>c_{1}(1+i) + c_{2}(1-i) = 0,</math> then <math>c_{1} = c_{2} = 0</math>. Rearranging the above equation, we obtain <math>(c_{1} + c_{2}) + (c_{1} - c_{2}) i = 0.</math> Now, a complex number is equal to <math>0</math> if and only if its real and imaginary parts are both <math>0</math>. So in this case, we conclude that <math>c_{1} + c_{2} = 0 \text{ and } c_{1} - c_{2} = 0.</math> This implies <math>c_{1} = c_{2}</math>, so that <math>c_{1} + c_{2} = 2c_{2} = 0</math>, which yields <math>c_{1} = c_{2} = 0</math>. Thus we conclude the vectors <math>1+i,1-i</math> are linearly independent in <math>\mathbb{C}</math> (over <math>\mathbb{R}</math>). | ||
| + | |} | ||
| Line 9: | Line 12: | ||
Show that <math>\{1+i,1-i \}</math> form a linearly independent set of vectors in <math>\mathbb{C}</math>, viewed as a vector space over <math>\mathbb{R}</math>. | Show that <math>\{1+i,1-i \}</math> form a linearly independent set of vectors in <math>\mathbb{C}</math>, viewed as a vector space over <math>\mathbb{R}</math>. | ||
| − | + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | |
| − | Recall that a set of vectors <math>\{v_{1},\ldots,v_{n}\}</math> in a vector space <math>V</math> (over a field <math>\mathbb{F}</math>) is said to be | + | !Proof: |
| + | |- | ||
| + | |Recall that a set of vectors <math>\{v_{1},\ldots,v_{n}\}</math> in a vector space <math>V</math> (over a field <math>\mathbb{F}</math>) is said to be ''linearly dependent'' if they are not linearly independent. More concretely, these vectors are linearly dependent if we can find scalars <math>c_{1},\ldots, c_{n} \in \mathbb{F}</math> ''not all equal to zero'' such that <math>c_{1}v_{1} + \cdots + c_{n}v_{n} = 0.</math> | ||
So for this problem, to show that <math>1+i</math> and <math>1-i</math> are not linearly dependent over <math>\mathbb{C}</math>, all we need to do is exhibit two complex scalars <math>c_{1}</math> and <math>c_{2}</math> that are not ''both'' zero such that <math>c_{1}(1+i) + c_{2}(1-i) = 0.</math> There are many choices for <math>c_{1}</math> and <math>c_{2}</math>, but one such example is <math>c_{1} = i</math> and <math>c_{2} = 1</math>. | So for this problem, to show that <math>1+i</math> and <math>1-i</math> are not linearly dependent over <math>\mathbb{C}</math>, all we need to do is exhibit two complex scalars <math>c_{1}</math> and <math>c_{2}</math> that are not ''both'' zero such that <math>c_{1}(1+i) + c_{2}(1-i) = 0.</math> There are many choices for <math>c_{1}</math> and <math>c_{2}</math>, but one such example is <math>c_{1} = i</math> and <math>c_{2} = 1</math>. | ||
| + | |} | ||
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Let <math>V</math> be a vector space over a field <math>\mathbb{F}</math>. If <math>\{v_{1},v_{2},v_{3},v_{4}\} \subseteq V</math> are a linearly independent set of vectors, then show that <math>\{v_{1} - v_{2}, v_{2} - v_{3}, v_{3} - v_{4},v_{4}\}</math> also form a linearly independent set of vectors in <math>V</math>. | Let <math>V</math> be a vector space over a field <math>\mathbb{F}</math>. If <math>\{v_{1},v_{2},v_{3},v_{4}\} \subseteq V</math> are a linearly independent set of vectors, then show that <math>\{v_{1} - v_{2}, v_{2} - v_{3}, v_{3} - v_{4},v_{4}\}</math> also form a linearly independent set of vectors in <math>V</math>. | ||
| − | + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | |
| − | Recall that the set of vectors <math>\{w_{1},\ldots, w_{n} \}</math> in a vector space <math>V</math> (over a field <math>\mathbb{F}</math>) are said to be '''linearly independent''' if whenever <math>c_{1},\ldots,c_{n}</math> are scalars in <math>\mathbb{F}</math> such that <math>c_{1}w_{1} + \cdots + c_{n}w_{n} = 0,</math> then <math>c_{1} = \cdots = c_{n} = 0</math>. | + | !Proof: |
| + | |- | ||
| + | |Recall that the set of vectors <math>\{w_{1},\ldots, w_{n} \}</math> in a vector space <math>V</math> (over a field <math>\mathbb{F}</math>) are said to be '''linearly independent''' if whenever <math>c_{1},\ldots,c_{n}</math> are scalars in <math>\mathbb{F}</math> such that <math>c_{1}w_{1} + \cdots + c_{n}w_{n} = 0,</math> then <math>c_{1} = \cdots = c_{n} = 0</math>. | ||
| + | So for this problem, we must show that whenever <math>c_{1},c_{2},c_{3},c_{4} \in \mathbb{F}</math> and <math>c_{1}(v_{1} - v_{2}) + c_{2}(v_{2} - v_{3}) + c_{3}(v_{3} - v_{4}) + c_{4}v_{4} = 0,</math> we have that <math>c_{1} = c_{2} = c_{3} = c_{4} = 0.</math> After rearranging terms in the above equation, we have that <math>c_{1}v_{1} + (c_{2} - c_{1})v_{2} + (c_{3} - c_{2})v_{3} + (c_{4} - c_{3})v_{4} = 0.</math> Now since the vectors <math>\{v_{1},v_{2},v_{3},v_{4}\}</math> are linearly independent in <math>V</math> by assumption, we have that | ||
| + | <math>c_{1} = 0 </math> | ||
| + | |||
| + | <math>c_{2} - c_{1} = 0 </math> | ||
| + | |||
| + | <math>c_{3} - c_{2} = 0 </math> | ||
| + | |||
| + | <math>c_{4} - c_{3} = 0.</math> | ||
| + | |||
| + | In other words, <math>c_{1} = c_{2} = c_{3} = c_{4} = 0</math>, so that <math>\{v_{1} - v_{2}, v_{2} - v_{3}, v_{3} - v_{4},v_{4}\}</math> form a linearly independent set as desired. | ||
| + | |} | ||
| + | |||
| + | |||
| + | '''Exercise''' | ||
| + | Prove that a vector space <math>V</math> over a field <math>\mathbb{F}</math> is infinite-dimensional if and only if there is a sequence <math>v_{1},v_{2},\ldots</math> in <math>V</math> such that <math>v_{1},\ldots,v_{m}</math> is linearly independent for every <math>m \in \mathbb{N}</math>. | ||
| + | |||
| + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Proof: | ||
| + | |- | ||
| + | |Recall that a vector space <math>V</math> is said to be '''finite dimensional''' if it is spanned by a finite list of vectors <math>w_{1},\ldots,w_{m} \in V.</math> In other words, <math>V</math> has finite dimension if every vector in <math>V</math> may be written as a linear combination of some list of vectors <math>w_{1},\ldots, w_{m} \in V</math>. On the other hand, a vector space <math>V</math> is '''infinite dimensional''' if it is not finite dimensional, i.e., <math>V</math> cannot be spanned by a finite list of vectors. Now before we proceed in the proof, we will need the following fact: | ||
| + | |||
| + | ''Lemma'' | ||
| + | Suppose <math>V</math> is a vector space over a field <math>\mathbb{F}</math>, and <math>v_{1},\ldots,v_{n}</math> are vectors that span <math>V</math>. If <math>w_{1},\ldots,w_{m}</math> in <math>V</math> are linearly independent, then <math>m \leq n</math>. | ||
| + | |||
| + | We are ready now to proceed with the proof:<br /> | ||
| + | ’<math>\Rightarrow</math>’: Suppose that <math>V</math> is an infinite dimensional vector space. Then, in particular, <math>V \neq 0</math>, so that there is some <math>v_{1} \neq 0</math> in <math>V</math>. Then <math>v_{1}</math> is a linearly independent vector in <math>V</math>. By way of induction now, suppose that for some <math>k \geq 1</math>, we have produced vectors <math>v_{1},\ldots, v_{k} \in V</math> such that <math>v_{1},\ldots,v_{k}</math> are linearly independent. Since <math>V</math> is infinite-dimensional, it cannot be spanned by the (finite!) list of vectors <math>v_{1},\ldots,v_{k}</math>. Thus we have that there is some <math>v_{k+1} \in V</math> such that <math>v_{k+1} \neq c_{1}v_{1} + \cdots + c_{k}v_{k}, \text{ for any }c_{1},\ldots, c_{k} \in \mathbb{F}.</math> We claim that now that <math>v_{1},\ldots,v_{k},v_{k+1}</math> form a linearly independent set in <math>V</math>. To see this, suppose that <math>a_{1}v_{1} + \cdots + a_{k}v_{k} + a_{k+1}v_{k+1} = 0 \text{ for some } a_{1},\ldots,a_{k},a_{k+1} \in \mathbb{F}.</math> Now if <math>a_{k+1} \neq 0</math>, then we may re-write the above equation as <math>v_{k+1} = \Big( \frac{-a_{1}}{a_{k+1}} \Big)v_{1} + \cdots + \Big( \frac{-a_{k}}{a_{k+1}} \Big)v_{k},</math> contradicting the fact that <math>v_{k+1}</math> is not in the span of <math>v_{1},\ldots,v_{k}</math>. So we conclude <math>a_{k+1} = 0</math>, and thus we have that <math>a_{1}v_{1} + \cdots + a_{k}v_{k} + a_{k+1}v_{k+1} | ||
| + | = a_{1}v_{1} + \cdots + a_{k}v_{k} + (0)v_{k+1}</math> <math>= a_{1}v_{1} + \cdots + a_{k}v_{k} = 0.</math> Now by induction hypothesis, since <math>v_{1},\ldots,v_{k}</math> are linearly independent, we must have <math>a_{1},\ldots,a_{k}</math> are all zero. Thus we’ve shown that <math>v_{1},\ldots, v_{k},v_{k+1}</math> also form a linearly independent set, completing the induction. Thus we have constructed a sequence of vectors <math>\{v_{k}\}^{\infty}_{k =1}</math> in <math>V</math> so that <math>v_{1},\ldots, v_{m}</math> is linearly independent for each <math>m \in \mathbb{N}</math>. | ||
| + | |||
| + | ’<math>\Leftarrow</math>’: On the other hand, suppose that <math>V</math> contains a sequence of vectors <math>\{v_{k}\}^{\infty}_{k =1}</math> so that <math>v_{1},\ldots, v_{m}</math> is linearly independent for each <math>m \in \mathbb{N}</math>. By way of contradiction, let’s suppose <math>V</math> is not infinite dimensional, i.e. is finite dimensional. Then <math>V</math> can be spanned by a finite list of vectors <math>w_{1},\ldots, w_{n} \in V</math>.<br /> | ||
| + | Now, since <math>V</math> contains a linearly independent set of <math>V</math>. | ||
| + | |} | ||
| + | |||
| + | |||
| + | '''Exercise''' | ||
| + | Suppose that <math>U</math> and <math>W</math> are subspaces of a vector space <math>V</math>. Prove that <math>U \cup W</math> is a subspace of <math>V</math> if and only if <math>U \subseteq W</math> or <math>W \subseteq U</math>. | ||
| + | |||
| + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Proof: | ||
| + | |- | ||
| + | |Recall that a subset <math>A</math> of a vector space <math>V</math> is a '''subspace''' of <math>V</math> if <math>A</math> itself is a vector space with the same addition and scalar multiplication operations as <math>V</math>.<br /> | ||
| + | ’<math>\Rightarrow</math>’: Instead of proving that <math>U \cup W</math> is a subspace of <math>V</math> implies <math>U \subseteq W</math> or <math>W \subseteq U</math>, we’ll show the ''contrapositive'' of this statement. That is, if <math>U \not\subseteq W</math> and <math>W \not\subseteq U</math>, then <math>U \cup W</math> is not a subspace of <math>V</math>. So suppose there is some <math>x \in U</math> that is not in <math>W</math>, and likewise that there is some <math>y \in W</math> that is not in <math>U</math>. We claim that <math>x + y \notin U \cup W</math>. For if it were, then <math>x+y</math> would lie in either <math>U</math> or <math>W</math>. If <math>x + y \in U</math>, then since <math>U</math> is a subspace, this would imply <math>y = (x+y) - x \in U,</math> contradicting our choice of <math>y</math>. Likewise, if <math>x + y \in W</math>, this would yield <math>x \in W</math>, which is again a contradiction. So we conclude that <math>x + y \notin U \cup W</math>, and thus <math>U \cup W</math> fails to be closed under addition, so cannot be a subspace of <math>V</math>.<br /> | ||
| + | ’<math>\Leftarrow</math>’: Suppose now that <math>U \subseteq W</math> or <math>W \subseteq U</math>. Then <math>U \cup W</math> is equal to either <math>W</math> or <math>U</math> respectively, which, by assumption are subspaces of <math>V</math>. | ||
| + | |} | ||
| + | |||
| + | |||
| + | |||
| + | Before we begin the next exercise, we will need the following notation: for an arbitrary non-empty set <math>X</math>, let <math>\mathbb{R}^{X}</math> denote the set of all functions <math>f \colon X \to \mathbb{R}</math>. Then <math>\mathbb{R}^{X}</math> is always a vector space, with addition and scalar multiplication defined pointwise. | ||
'''Exercise''' | '''Exercise''' | ||
| − | + | Let <math>b \in \mathbb{R}</math> and consider the set <math>W = \Big\{ f \in \mathbb{R}^{[0,1]} \colon f \text{ is continuous and} \int^{1}_{0} f(x) dx = b \Big\}.</math> Show that <math>W</math> is a subspace of <math>\mathbb{R}^{[0,1]}</math> if and only if <math>b = 0</math>. | |
| − | + | ||
| − | + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | |
| − | + | !Proof: | |
| − | + | |- | |
| − | </math> | + | |Recall that a subset <math>A</math> of a vector space <math>V</math> is a '''subspace''' of <math>V</math> if <math>A</math> itself is a vector space with the same addition and scalar multiplication operations as <math>V</math>. There is a very convenient test that determines if <math>A</math> is a subspace of <math>V</math>, sometimes called the ''subspace test''. It says the following: |
| + | |||
| + | ('''Subspace Test''') Suppose that <math>A \subseteq V</math>, where <math>V</math> is a vector space over a field <math>\mathbb{F}</math>. Then <math>A</math> is a subspace of <math>V</math> if and only if the following conditions are met: | ||
| + | |||
| + | # <math>0_{V} \in A</math>, | ||
| + | # <math>c \in \mathbb{F}, v \in A \Rightarrow cv \in A</math> | ||
| + | # <math>v,w \in A \Rightarrow v + w \in A</math>. | ||
| + | |||
| + | We are now ready to proceed with the proof: | ||
| + | |||
| + | ’<math>\Rightarrow</math>’: Suppose <math>W</math> is a subspace of <math>\mathbb{R}^{[0,1]}</math>. Then by condition <math>(1)</math> of the subspace test, <math>W</math> contains the zero vector of <math>\mathbb{R}^{[0,1]}</math>, which is just the function that maps <math>x</math> to <math>0</math> for all <math>x \in [0,1]</math>. We will write this zero vector as <math>\textbf{0}</math>. Now since <math>\textbf{0} \in W</math>, by definition of being in <math>W</math>, we must have that <math>\int^{1}_{0} \textbf{0}(x)dx = b.</math> On the other hand, when we actually integrate <math>\textbf{0}</math>, we find the integral must be zero. Thus <math>b = 0</math> as desired.<br /> | ||
| + | ’<math>\Leftarrow</math>’: Say <math>b = 0</math>. We will show that <math>W</math> is a subspace of <math>V</math> by showing that it passes all three conditions of the subspace test above. For condition (1), just note that by our previous remark, <math>\int^{1}_{0} \textbf{0}(x)dx = 0 = b</math>, and since <math>\textbf{0}</math> is continuous, we have that <math>\textbf{0} \in W</math>. For condition (2), suppose that <math>c \in \mathbb{R}</math> and <math>f \in W</math>. We must show that <math>cf \in W</math>. Since a continuous function multiplied by a constant is still continuous, <math>cf</math> is still a continuous function. Now, <math>\int^{1}_{0}(cf)(x)dx = \int^{1}_{0}c[f(x)]dx = c\int^{1}_{0} f(x)dx = c(0) = 0,</math> so that we conclude <math>cf \in W</math>. Lastly, for condition (3), we must show that if <math>f,g \in W</math>, then <math>f + g \in W</math>. The addition of two continuous functions is always continuous, so that <math>f + g</math> is continuous. Now since <math>f,g \in W</math>, we have that <math>\int^{1}_{0} (f + g)(x)dx = \int^{1}_{0}[f(x) + g(x)]dx = \int^{1}_{0}f(x)dx + \int^{1}_{0}g(x)dx = 0 + 0 = 0,</math> so that <math>f + g \in W</math>, and thus <math>W</math> satisfies all three conditions of the subspace test. | ||
| + | |} | ||
| + | |||
| + | |||
| + | '''Exercise''' | ||
| + | Prove or give a counterexample to the following statement: If <math>U_{1},U_{2},W</math> are subspaces of a vector space <math>V</math> with <math>V = U_{1} \oplus W \text { and } V = U_{2} \oplus W,</math> then <math>U_{1} = U_{2}</math>. | ||
| + | |||
| + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Proof: | ||
| + | |- | ||
| + | |Let’s think about what it means for two subspaces <math>A,B</math> of a vector space <math>C</math> to satisfy <math>C = A \oplus B</math>. This means that <math>A \cap B = \{0_{C}\}</math> and that <math>C = A + B</math>. In other words, for any <math>c \in C</math>, we may write <math>c</math> uniquely in the form <math>c = a + b</math>, where <math>a \in A</math> and <math>b \in B</math>.<br /> | ||
| + | It turns out the statement of the problem is '''false''', so that we must provide a counterexample to this statement: Let <math>V = \mathbb{R}^{2}</math> and consider its subspaces (one should check that they actually form subspaces first): | ||
| + | <math>W = \{ (x,0) \in \mathbb{R}^{2} \colon x \in \mathbb{R} \} </math> | ||
| + | |||
| + | <math>U_{1} = \{ (0,y) \in \mathbb{R}^{2} \colon y \in \mathbb{R} \} </math> | ||
| + | |||
| + | <math>U_{2} = \{ (z,z) \in \mathbb{R}^{2} \colon z \in \mathbb{R} \} </math> | ||
| + | |||
| + | Then <math>U_{1}</math> and <math>U_{2}</math> are not the same subspaces of <math>V</math>, so that all we need to check is <math>U_{1} \oplus W = V = U_{2} \oplus W.</math> Suppose that <math>(x,y) \in V</math>. Then <math>(x,y) = (x,0) + (0,y)</math>, where <math>(x,0) \in W</math> and <math>(0,y) \in U_{1}</math>, so that <math>V = W + U_{1}</math>. Now if <math>(a,b) \in W \cap U_{1}</math>, then <math>(a,b) = (x,0) \in W</math> for some <math>x \in \mathbb{R}</math>, and thus <math>b = 0</math>. Likewise, <math>(a,b) = (0,y) \in W</math> for some <math>y \in \mathbb{R}</math>, so that <math>a = 0</math>. This shows <math>W \cap U_{1} = \{(0,0)\}</math>, and hence <math>V = U_{1} \oplus W</math>.<br /> | ||
| + | Now again say <math>(x,y) \in V</math>. Then <math>(x,y) = (x-y,0) + (y,y)</math>, where <math>(x-y,0) \in W</math> and <math>(y,y) \in U_{2}</math> so that <math>V = W + U_{2}</math>. Now suppose <math>(a,b) \in W \cap U_{2}</math>. Then <math>(a,b) = (x,0) \in W</math> for some <math>x \in \mathbb{R}</math>, so that <math>b = 0</math>. Likewise, <math>(a,b) = (a,0) = (z,z) \in U_{2}</math> for some <math>z \in \mathbb{R}</math>, thus <math>z = 0 = a</math>, so that we conclude <math>W \cap U_{2} = \{(0,0)\}</math>, and thus <math>V = U_{2} \oplus W</math>. | ||
| + | |} | ||
Latest revision as of 00:09, 16 November 2015
Exercise Show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1+i,1-i \}} form a linearly independent set of vectors in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} , viewed as a vector space over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} .
| Proof: |
|---|
| Recall that the set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{1},\ldots, v_{n} \}} in a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} (over a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}} ) are said to be linearly independent if whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1},\ldots,c_{n}} are scalars in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}v_{1} + \cdots + c_{n}v_{n} = 0,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = \cdots = c_{n} = 0} . So for this problem, since we’re considering the complex numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} as a vector space over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} , we must show that whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1},c_{2} \in \mathbb{R}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}(1+i) + c_{2}(1-i) = 0,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = c_{2} = 0} . Rearranging the above equation, we obtain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (c_{1} + c_{2}) + (c_{1} - c_{2}) i = 0.} Now, a complex number is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} if and only if its real and imaginary parts are both Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} . So in this case, we conclude that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} + c_{2} = 0 \text{ and } c_{1} - c_{2} = 0.} This implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = c_{2}} , so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} + c_{2} = 2c_{2} = 0} , which yields Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = c_{2} = 0} . Thus we conclude the vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+i,1-i} are linearly independent in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} (over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} ). |
Exercise
Show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1+i,1-i \}}
form a linearly independent set of vectors in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}}
, viewed as a vector space over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}}
.
| Proof: |
|---|
| Recall that a set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{1},\ldots,v_{n}\}}
in a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
(over a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}}
) is said to be linearly dependent if they are not linearly independent. More concretely, these vectors are linearly dependent if we can find scalars Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1},\ldots, c_{n} \in \mathbb{F}}
not all equal to zero such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}v_{1} + \cdots + c_{n}v_{n} = 0.}
So for this problem, to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-i} are not linearly dependent over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} , all we need to do is exhibit two complex scalars Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{2}} that are not both zero such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}(1+i) + c_{2}(1-i) = 0.} There are many choices for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{2}} , but one such example is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{2} = 1} . |
Exercise
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
be a vector space over a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}}
. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{1},v_{2},v_{3},v_{4}\} \subseteq V}
are a linearly independent set of vectors, then show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{1} - v_{2}, v_{2} - v_{3}, v_{3} - v_{4},v_{4}\}}
also form a linearly independent set of vectors in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
.
| Proof: |
|---|
| Recall that the set of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{w_{1},\ldots, w_{n} \}}
in a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
(over a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}}
) are said to be linearly independent if whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1},\ldots,c_{n}}
are scalars in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}}
such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}w_{1} + \cdots + c_{n}w_{n} = 0,}
then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = \cdots = c_{n} = 0}
.
So for this problem, we must show that whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1},c_{2},c_{3},c_{4} \in \mathbb{F}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}(v_{1} - v_{2}) + c_{2}(v_{2} - v_{3}) + c_{3}(v_{3} - v_{4}) + c_{4}v_{4} = 0,} we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = c_{2} = c_{3} = c_{4} = 0.} After rearranging terms in the above equation, we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1}v_{1} + (c_{2} - c_{1})v_{2} + (c_{3} - c_{2})v_{3} + (c_{4} - c_{3})v_{4} = 0.} Now since the vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{1},v_{2},v_{3},v_{4}\}} are linearly independent in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} by assumption, we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = 0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{2} - c_{1} = 0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{3} - c_{2} = 0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{4} - c_{3} = 0.} In other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1} = c_{2} = c_{3} = c_{4} = 0} , so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{1} - v_{2}, v_{2} - v_{3}, v_{3} - v_{4},v_{4}\}} form a linearly independent set as desired. |
Exercise
Prove that a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
over a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}}
is infinite-dimensional if and only if there is a sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},v_{2},\ldots}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots,v_{m}}
is linearly independent for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{N}}
.
| Proof: |
|---|
| Recall that a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
is said to be finite dimensional if it is spanned by a finite list of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{1},\ldots,w_{m} \in V.}
In other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
has finite dimension if every vector in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
may be written as a linear combination of some list of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{1},\ldots, w_{m} \in V}
. On the other hand, a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
is infinite dimensional if it is not finite dimensional, i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
cannot be spanned by a finite list of vectors. Now before we proceed in the proof, we will need the following fact:
Lemma Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a vector space over a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots,v_{n}} are vectors that span Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{1},\ldots,w_{m}} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} are linearly independent, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \leq n} . We are ready now to proceed with the proof: ’Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow}
’: On the other hand, suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
contains a sequence of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_{k}\}^{\infty}_{k =1}}
so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{1},\ldots, v_{m}}
is linearly independent for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{N}}
. By way of contradiction, let’s suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
is not infinite dimensional, i.e. is finite dimensional. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
can be spanned by a finite list of vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{1},\ldots, w_{n} \in V}
. |
Exercise
Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}
are subspaces of a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
. Prove that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \cup W}
is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \subseteq W}
or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W \subseteq U}
.
| Proof: |
|---|
| Recall that a subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
of a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
itself is a vector space with the same addition and scalar multiplication operations as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
. ’Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow}
’: Instead of proving that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \cup W}
is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \subseteq W}
or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W \subseteq U}
, we’ll show the contrapositive of this statement. That is, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \not\subseteq W}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W \not\subseteq U}
, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \cup W}
is not a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
. So suppose there is some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in U}
that is not in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}
, and likewise that there is some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in W}
that is not in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U}
. We claim that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y \notin U \cup W}
. For if it were, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y}
would lie in either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U}
or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}
. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y \in U}
, then since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U}
is a subspace, this would imply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = (x+y) - x \in U,}
contradicting our choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y}
. Likewise, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y \in W}
, this would yield Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in W}
, which is again a contradiction. So we conclude that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y \notin U \cup W}
, and thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \cup W}
fails to be closed under addition, so cannot be a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
. |
Before we begin the next exercise, we will need the following notation: for an arbitrary non-empty set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{X}} denote the set of all functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \colon X \to \mathbb{R}} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{X}} is always a vector space, with addition and scalar multiplication defined pointwise.
Exercise Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \in \mathbb{R}} and consider the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = \Big\{ f \in \mathbb{R}^{[0,1]} \colon f \text{ is continuous and} \int^{1}_{0} f(x) dx = b \Big\}.} Show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{[0,1]}} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 0} .
| Proof: |
|---|
| Recall that a subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
of a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
itself is a vector space with the same addition and scalar multiplication operations as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
. There is a very convenient test that determines if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
, sometimes called the subspace test. It says the following:
(Subspace Test) Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subseteq V} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a vector space over a field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} if and only if the following conditions are met:
We are now ready to proceed with the proof: ’Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow}
’: Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}
is a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{[0,1]}}
. Then by condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1)}
of the subspace test, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}
contains the zero vector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^{[0,1]}}
, which is just the function that maps Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}
for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in [0,1]}
. We will write this zero vector as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{0}}
. Now since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{0} \in W}
, by definition of being in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}
, we must have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int^{1}_{0} \textbf{0}(x)dx = b.}
On the other hand, when we actually integrate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{0}}
, we find the integral must be zero. Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 0}
as desired. |
Exercise
Prove or give a counterexample to the following statement: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{1},U_{2},W}
are subspaces of a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = U_{1} \oplus W \text { and } V = U_{2} \oplus W,}
then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{1} = U_{2}}
.
| Proof: |
|---|
| Let’s think about what it means for two subspaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A,B}
of a vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}
to satisfy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = A \oplus B}
. This means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \cap B = \{0_{C}\}}
and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = A + B}
. In other words, for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \in C}
, we may write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}
uniquely in the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = a + b}
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in A}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \in B}
. It turns out the statement of the problem is false, so that we must provide a counterexample to this statement: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = \mathbb{R}^{2}} and consider its subspaces (one should check that they actually form subspaces first): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = \{ (x,0) \in \mathbb{R}^{2} \colon x \in \mathbb{R} \} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{1} = \{ (0,y) \in \mathbb{R}^{2} \colon y \in \mathbb{R} \} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{2} = \{ (z,z) \in \mathbb{R}^{2} \colon z \in \mathbb{R} \} } Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{1}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{2}}
are not the same subspaces of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}
, so that all we need to check is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{1} \oplus W = V = U_{2} \oplus W.}
Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y) \in V}
. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y) = (x,0) + (0,y)}
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,0) \in W}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,y) \in U_{1}}
, so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = W + U_{1}}
. Now if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b) \in W \cap U_{1}}
, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b) = (x,0) \in W}
for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{R}}
, and thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 0}
. Likewise, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b) = (0,y) \in W}
for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in \mathbb{R}}
, so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 0}
. This shows Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W \cap U_{1} = \{(0,0)\}}
, and hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = U_{1} \oplus W}
. |