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>). | ||
| + | |} | ||
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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 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>. | ||
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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 | 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 | ||
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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. | 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. | ||
| + | |} | ||
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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>. | 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;" | |
| − | 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: | + | !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'' | ''Lemma'' | ||
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’<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 /> | ’<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>. | Now, since <math>V</math> contains a linearly independent set of <math>V</math>. | ||
| + | |} | ||
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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>. | 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;" | |
| − | 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 /> | + | !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>\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>. | ’<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>. | ||
| + | |} | ||
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| − | Before we begin, 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. | + | 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>. | 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;" | |
| − | 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: | + | !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>. 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: | ('''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: | ||
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’<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>\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. | ’<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. | ||
| + | |} | ||
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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>. | 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;" | |
| − | 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 /> | + | !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): | 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>W = \{ (x,0) \in \mathbb{R}^{2} \colon x \in \mathbb{R} \} </math> | ||
| Line 98: | Line 115: | ||
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 /> | 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>. | 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 form a linearly independent set of vectors in , viewed as a vector space over .
| Proof: |
|---|
| Recall that the set of vectors Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{v_{1},\ldots ,v_{n}\}} in a vector space (over a field ) are said to be linearly independent if whenever are scalars in such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{1}v_{1}+\cdots +c_{n}v_{n}=0,} then . So for this problem, since we’re considering the complex numbers as a vector space over , we must show that whenever Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{1},c_{2}\in \mathbb {R} } and then . Rearranging the above equation, we obtain Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (c_{1}+c_{2})+(c_{1}-c_{2})i=0.} Now, a complex number is equal to if and only if its real and imaginary parts are both . So in this case, we conclude that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{1}+c_{2}=0{\text{ and }}c_{1}-c_{2}=0.} This implies Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{1}=c_{2}} , so that , which yields . Thus we conclude the vectors Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1+i,1-i} are linearly independent in (over ). |
Exercise
Show that form a linearly independent set of vectors in , viewed as a vector space over .
| Proof: |
|---|
| Recall that a set of vectors Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{v_{1},\ldots ,v_{n}\}}
in a vector space (over a field ) 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{1},\ldots ,c_{n}\in \mathbb {F} }
not all equal to zero such that
So for this problem, to show that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1+i} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1-i} are not linearly dependent over , all we need to do is exhibit two complex scalars and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{2}} that are not both zero such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{1}(1+i)+c_{2}(1-i)=0.} There are many choices for and , but one such example is and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{2}=1} . |
Exercise
Let be a vector space over a field . If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{v_{1},v_{2},v_{3},v_{4}\}\subseteq V}
are a linearly independent set of vectors, then show that also form a linearly independent set of vectors in .
| Proof: |
|---|
| Recall that the set of vectors Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{w_{1},\ldots ,w_{n}\}}
in a vector space (over a field ) are said to be linearly independent if whenever are scalars in 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 .
So for this problem, we must show that whenever Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 , 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}
. |