Difference between revisions of "Section 1.4 Homework"

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(Created page with "'''1.''' Find a subset <math>C \subset \mathbb{F}^2</math> that is closed under scalar multiplication but not under addition of vectors.<br /> <br /> ''Solution'' There are m...")
 
 
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''Solution''
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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There are many possible answers that work. Here is one of them.<br />
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!Solution:
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<math>\mathbb{F}^2 = \mathbb{R}^2</math> with <math>C = \{(x,y): x^2 = y^2\}</math>. Then if <math>(x,y)\in C</math> and <math>\alpha \in \mathbb{R}</math> we have<br />
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|There are many possible answers that work. Here is one of them.<br /> <math>\mathbb{F}^2 = \mathbb{R}^2</math> with <math>C = \{(x,y): x^2 = y^2\}</math>. Then if <math>(x,y)\in C</math> and <math>\alpha \in \mathbb{R}</math> we have
 
<math>\alpha (x,y) = (\alpha x, \alpha y) \in C</math> because <math>x^2 = y^2 \Rightarrow (\alpha x)^2 = \alpha^2 x^2 = \alpha^2 y^2 = (\alpha y)^2</math> so that <math>C</math> is closed under scalar multiplication. However, <math>(1,1) \in C</math> and <math>(1,-1) \in C</math>, but <math>(1,1)+(1,-1) = (2,0) \notin C</math> so that <math>C</math> is not closed under addition of vectors.
 
<math>\alpha (x,y) = (\alpha x, \alpha y) \in C</math> because <math>x^2 = y^2 \Rightarrow (\alpha x)^2 = \alpha^2 x^2 = \alpha^2 y^2 = (\alpha y)^2</math> so that <math>C</math> is closed under scalar multiplication. However, <math>(1,1) \in C</math> and <math>(1,-1) \in C</math>, but <math>(1,1)+(1,-1) = (2,0) \notin C</math> so that <math>C</math> is not closed under addition of vectors.
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'''2.''' Find a subset <math>A \subset \mathbb{C}^2</math> that is closed under vector addition but not under multiplication by complex number.<br />
 
'''2.''' Find a subset <math>A \subset \mathbb{C}^2</math> that is closed under vector addition but not under multiplication by complex number.<br />
  
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''Solution''
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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Many possible answers again. Here are a few:<br />
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!Solution:
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|Many possible answers again. Here are a few:<br />
 
<math>A = \mathbb{N}^2</math>, <math>A = \mathbb{Z}^2</math>, <math>A = \mathbb{Q}^2</math>, <math>A=\mathbb{R}^2</math>, all are closed under addition, but if you multiply <math>(1,1) \in A</math> for all of these <math>A</math> by <math>i</math> then you get <math>(i,i) \notin A</math>.
 
<math>A = \mathbb{N}^2</math>, <math>A = \mathbb{Z}^2</math>, <math>A = \mathbb{Q}^2</math>, <math>A=\mathbb{R}^2</math>, all are closed under addition, but if you multiply <math>(1,1) \in A</math> for all of these <math>A</math> by <math>i</math> then you get <math>(i,i) \notin A</math>.
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''Solution''
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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Here them using <math>Q</math> can be a hint. If you let <math>Q = \mathbb{Q}</math>, the rational numbers, then they will be closed under addition, but not scalar multiplication. That is because <math>1 \in Q</math> and <math>\sqrt{2} \in \mathbb{R}</math>, but <math>\sqrt{2} = \sqrt{2} \cdot 1 \notin Q</math>.<br />
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!Solution:
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|Here them using <math>Q</math> can be a hint. If you let <math>Q = \mathbb{Q}</math>, the rational numbers, then they will be closed under addition, but not scalar multiplication. That is because <math>1 \in Q</math> and <math>\sqrt{2} \in \mathbb{R}</math>, but <math>\sqrt{2} = \sqrt{2} \cdot 1 \notin Q</math>.<br />
 
Another possible answer is <math>Q = \{ x \in \mathbb{R}: x > 0\}</math>. Then this will be closed under addition since the sum of two positive numbers is still positive, but <math>1 \in Q</math> and <math>-1 \in \mathbb{R}</math> and <math>-1 = -1 \cdot 1 \notin Q</math>.
 
Another possible answer is <math>Q = \{ x \in \mathbb{R}: x > 0\}</math>. Then this will be closed under addition since the sum of two positive numbers is still positive, but <math>1 \in Q</math> and <math>-1 \in \mathbb{R}</math> and <math>-1 = -1 \cdot 1 \notin Q</math>.
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''Solution''
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To begin we don’t need to show any of the first four axioms are true as they only involve addition of vectors and since <math>V^*</math> has the same additive structure as <math>V</math> and <math>V</math> is a vector space, the first four axioms will still be true. For the remaining four properties we simply check that they will hold.<br />
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!Solution:
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|To begin we don’t need to show any of the first four axioms are true as they only involve addition of vectors and since <math>V^*</math> has the same additive structure as <math>V</math> and <math>V</math> is a vector space, the first four axioms will still be true. For the remaining four properties we simply check that they will hold.<br />
 
5) <math>1 * x = \bar{1} x = 1x = x</math><br />
 
5) <math>1 * x = \bar{1} x = 1x = x</math><br />
 
6) <math>\alpha * (\beta * x) = \alpha * (\bar{\beta} x) = \bar{\alpha} \bar{ \beta} x = \overline{ \alpha \beta} x = (\alpha \beta) * x</math><br />
 
6) <math>\alpha * (\beta * x) = \alpha * (\bar{\beta} x) = \bar{\alpha} \bar{ \beta} x = \overline{ \alpha \beta} x = (\alpha \beta) * x</math><br />
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8) <math>(\alpha + \beta) * x = \overline{\alpha + \beta} x + (\bar{\alpha} + \bar{\beta}) x = \bar{\alpha}x + \bar{\beta} x = \alpha * x + \beta * x</math><br />
 
8) <math>(\alpha + \beta) * x = \overline{\alpha + \beta} x + (\bar{\alpha} + \bar{\beta}) x = \bar{\alpha}x + \bar{\beta} x = \alpha * x + \beta * x</math><br />
 
Therefore <math>V^*</math> is a complex vector space.
 
Therefore <math>V^*</math> is a complex vector space.
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'''7.''' Let <math>P_n</math> be the set of polynomials in <math>\mathbb{F}[t]</math> of degree <math>\leq n</math>.<br />
 
'''7.''' Let <math>P_n</math> be the set of polynomials in <math>\mathbb{F}[t]</math> of degree <math>\leq n</math>.<br />
 
(a) Show that <math>P_n</math> is a vector space.<br />
 
(a) Show that <math>P_n</math> is a vector space.<br />
 
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<br />
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''Solution''
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Suppose <math>x,y \in P_n</math> and <math>\alpha \in \mathbb{R}</math>. Then <math>x = a_n t^n + \cdots + a_0</math> and <math>y = b_n t^n + \cdots + b_0</math>. So that <math>x+y = (a_n+b_n)t^n + (a_0 + b_0) \in P_n</math>. Also <math>\alpha x = (\alpha a_n)t^n + \cdots \alpha a_0 \in P_n</math> so that <math>P_n</math> is closed under addition and scalar multiplication. Also <math>x + y = (a_n+b_n)t^n + (a_0 + b_0) = (b_n+a_n)t^n + (b_0 + a_0) = y + x</math> so addition is commutative. Similarly for associative. The zero vector in <math>P_n</math> is the zero polynomial with all coefficients equal to 0. The additive inverse of <math>x</math> is <math>(-x) = (-a_n)t^n + (-a_0)</math>. Also, <math>(1) x = x</math>, <math>\alpha(\beta x) = (\alpha \beta) x</math>, <math>(\alpha + \beta) x = \alpha x + \beta x</math> and <math>\alpha (x + y) = \alpha x + \alpha y</math> just by writing each of them out. Therefore <math>P_n</math> is a vector space.<br />
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!Solution:
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|Suppose <math>x,y \in P_n</math> and <math>\alpha \in \mathbb{R}</math>. Then <math>x = a_n t^n + \cdots + a_0</math> and <math>y = b_n t^n + \cdots + b_0</math>. So that <math>x+y = (a_n+b_n)t^n + (a_0 + b_0) \in P_n</math>. Also <math>\alpha x = (\alpha a_n)t^n + \cdots \alpha a_0 \in P_n</math> so that <math>P_n</math> is closed under addition and scalar multiplication. Also <math>x + y = (a_n+b_n)t^n + (a_0 + b_0) = (b_n+a_n)t^n + (b_0 + a_0) = y + x</math> so addition is commutative. Similarly for associative. The zero vector in <math>P_n</math> is the zero polynomial with all coefficients equal to 0. The additive inverse of <math>x</math> is <math>(-x) = (-a_n)t^n + (-a_0)</math>. Also, <math>(1) x = x</math>, <math>\alpha(\beta x) = (\alpha \beta) x</math>, <math>(\alpha + \beta) x = \alpha x + \beta x</math> and <math>\alpha (x + y) = \alpha x + \alpha y</math> just by writing each of them out. Therefore <math>P_n</math> is a vector space.<br />
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(b) Show that the space of polynomials of degree <math>n \geq 1</math> is <math>P_n - P_{n-1}</math> and does not form a subspace.<br />
 
(b) Show that the space of polynomials of degree <math>n \geq 1</math> is <math>P_n - P_{n-1}</math> and does not form a subspace.<br />
 
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''Solution''
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First off, the space is equal to <math>P_n - P_{n-1}</math> because the polynomials that have exactly degree <math>n</math> are in <math>P_n</math> but not in <math>P_{n-1}</math> and there are no other polynomials in <math>P_n</math> that aren’t also in <math>P_{n-1}</math>. Now this set does not form a subspace because it is not closed under addition. <math>p(t) = t^n</math> and <math>q(t) = -t^n + 1</math> are both polynomials of degree exactly <math>n</math>. However, <math>p+q = 1</math> is a polynomial of degree 0 not <math>n</math>.<br />
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!Solution:
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|First off, the space is equal to <math>P_n - P_{n-1}</math> because the polynomials that have exactly degree <math>n</math> are in <math>P_n</math> but not in <math>P_{n-1}</math> and there are no other polynomials in <math>P_n</math> that aren’t also in <math>P_{n-1}</math>. Now this set does not form a subspace because it is not closed under addition. <math>p(t) = t^n</math> and <math>q(t) = -t^n + 1</math> are both polynomials of degree exactly <math>n</math>. However, <math>p+q = 1</math> is a polynomial of degree 0 not <math>n</math>.<br />
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(c) If <math>f(t): \mathbb{F} \to \mathbb{F}</math>, show that <math>V = \{ p(t)f(t): p \in P_n\}</math> is a subspace of <math>Func(\mathbb{F}, \mathbb{F})</math>.<br />
 
(c) If <math>f(t): \mathbb{F} \to \mathbb{F}</math>, show that <math>V = \{ p(t)f(t): p \in P_n\}</math> is a subspace of <math>Func(\mathbb{F}, \mathbb{F})</math>.<br />
 
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''Solution''
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To show it is a subspace, we only need to check that it is closed under addition and scalar multiplication. Let <math>x,y \in V</math> and <math>\alpha \in \mathbb{F}</math>. Then <math>x = p(t)f(t)</math> for some polynomial <math>p \in P_n</math> and <math>y = q(t)f(t)</math> for some polynomial <math>q \in P_n</math>. Then <math>x + y = p(t)f(t)+q(t)f(t) = (p(t)+q(t))f(t)</math>. But since <math>p(t)+q(t)</math> is just another polynomial in <math>P_n</math>, then <math>x+y</math> is exactly of the form polynomial times <math>f(t)</math>. Thus <math>x+y \in V</math>. Also <math>\alpha x = \alpha (p(t)f(t)) = (\alpha p(t)) f(t)</math> which is again of the form polynomial times <math>f(t)</math> so <math>\alpha x \in V</math>. Therefore <math>V</math> is a subspace. 8) <math>(\alpha + \beta) * x = \overline{\alpha + \beta} x + (\bar{\alpha} + \bar{\beta}) x = \bar{\alpha}x + \bar{\beta} x = \alpha * x + \beta * x</math><br />
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!Solution:
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Therefore <math>V^*</math> is a complex vector space.
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|To show it is a subspace, we only need to check that it is closed under addition and scalar multiplication. Let <math>x,y \in V</math> and <math>\alpha \in \mathbb{F}</math>. Then <math>x = p(t)f(t)</math> for some polynomial <math>p \in P_n</math> and <math>y = q(t)f(t)</math> for some polynomial <math>q \in P_n</math>. Then <math>x + y = p(t)f(t)+q(t)f(t) = (p(t)+q(t))f(t)</math>. But since <math>p(t)+q(t)</math> is just another polynomial in <math>P_n</math>, then <math>x+y</math> is exactly of the form polynomial times <math>f(t)</math>. Thus <math>x+y \in V</math>. Also <math>\alpha x = \alpha (p(t)f(t)) = (\alpha p(t)) f(t)</math> which is again of the form polynomial times <math>f(t)</math> so <math>\alpha x \in V</math>. Therefore <math>V</math> is a subspace. 8) <math>(\alpha + \beta) * x = \overline{\alpha + \beta} x + (\bar{\alpha} + \bar{\beta}) x = \bar{\alpha}x + \bar{\beta} x = \alpha * x + \beta * x</math>
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|Therefore <math>V^*</math> is a complex vector space.
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(a) Show that if we use <math>0_V = 1</math> and <math>-x = x^{-1}</math>, then the first four axioms for a vector space are satisfied.<br />
 
(a) Show that if we use <math>0_V = 1</math> and <math>-x = x^{-1}</math>, then the first four axioms for a vector space are satisfied.<br />
 
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''Solution''
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1) <math>x\boxplus y = xy = yx = y \boxplus x</math> so addition is commutative.<br />
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!Solution:
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|1) <math>x\boxplus y = xy = yx = y \boxplus x</math> so addition is commutative.<br />
 
2) <math>(x \boxplus y) \boxplus z = (xy) \boxplus z = (xy)z = xyz = x(yz) = x(y \boxplus z) = x\boxplus (y \boxplus z)</math> and addition is associative.<br />
 
2) <math>(x \boxplus y) \boxplus z = (xy) \boxplus z = (xy)z = xyz = x(yz) = x(y \boxplus z) = x\boxplus (y \boxplus z)</math> and addition is associative.<br />
 
3) <math>x \boxplus 0_V = x \boxplus 1 = 1x = x</math><br />
 
3) <math>x \boxplus 0_V = x \boxplus 1 = 1x = x</math><br />
 
4) <math>x \boxplus -x = x \boxplus x^{-1} = xx^{-1} = 1 = 0_V</math><br />
 
4) <math>x \boxplus -x = x \boxplus x^{-1} = xx^{-1} = 1 = 0_V</math><br />
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<br />
 
<br />
 
(b) Which of the scalar multiplication properties do not hold?<br />
 
(b) Which of the scalar multiplication properties do not hold?<br />
 
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''Solution''
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The only property that won’t hold is associativity of scalar multiplication.<br />
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!Solution:
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|The only property that won’t hold is associativity of scalar multiplication.<br />
 
<math>\alpha \boxdot (\beta \boxdot x) = \alpha (e^\beta x) = e^\alpha e^\beta x</math> which is not the same as <math>(\alpha \beta) \boxdot x = e^{\alpha \beta} x</math>.
 
<math>\alpha \boxdot (\beta \boxdot x) = \alpha (e^\beta x) = e^\alpha e^\beta x</math> which is not the same as <math>(\alpha \beta) \boxdot x = e^{\alpha \beta} x</math>.
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Latest revision as of 22:55, 15 November 2015

1. Find 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 C \subset \mathbb{F}^2} that is closed under scalar multiplication but not under addition of vectors.

Solution:
There are many possible answers that work. Here is one of them.
Failed to parse (MathML with SVG or PNG fallback (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}^2 = \mathbb{R}^2} 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 C = \{(x,y): x^2 = y^2\}} . Then 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 C} 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 \alpha \in \mathbb{R}} we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha (x,y) = (\alpha x, \alpha y) \in C} because Failed to parse (MathML with SVG or PNG fallback (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^2 = y^2 \Rightarrow (\alpha x)^2 = \alpha^2 x^2 = \alpha^2 y^2 = (\alpha y)^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} is closed under scalar multiplication. However, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,1) \in C} 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,-1) \in C} , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,1)+(1,-1) = (2,0) \notin C} 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} is not closed under addition of vectors.


2. Find 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 \subset \mathbb{C}^2} that is closed under vector addition but not under multiplication by complex number.

Solution:
Many possible answers again. Here are a few:

Failed to parse (MathML with SVG or PNG fallback (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 = \mathbb{N}^2} , Failed to parse (MathML with SVG or PNG fallback (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 = \mathbb{Z}^2} , Failed to parse (MathML with SVG or PNG fallback (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 = \mathbb{Q}^2} , Failed to parse (MathML with SVG or PNG fallback (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=\mathbb{R}^2} , all are closed under addition, but if you multiply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,1) \in A} for all of these Failed to parse (MathML with SVG or PNG fallback (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} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} then you get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (i,i) \notin A} .


3. Find 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 Q \subset \mathbb{R}} that is closed under addition but not scalar multiplication by real scalars.

Solution:
Here them using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} can be a hint. If you 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 Q = \mathbb{Q}} , the rational numbers, then they will be closed under addition, but not scalar multiplication. That is because Failed to parse (MathML with SVG or PNG fallback (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 \in Q} 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 \sqrt{2} \in \mathbb{R}} , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2} = \sqrt{2} \cdot 1 \notin Q} .

Another possible answer 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 Q = \{ x \in \mathbb{R}: x > 0\}} . Then this will be closed under addition since the sum of two positive numbers is still positive, but Failed to parse (MathML with SVG or PNG fallback (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 \in Q} 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 \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 -1 = -1 \cdot 1 \notin Q} .


6. 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 complex vector space i.e., a vector space where the scalars are Failed to parse (MathML with SVG or PNG fallback (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}} . Define Failed to parse (MathML with SVG or PNG fallback (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^*} as the complex vector space whose additive structure is that 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} but where complex scalar multiplication is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda * x = \bar{\lambda} x} . 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^*} is a complex vector space.

Solution:
To begin we don’t need to show any of the first four axioms are true as they only involve addition of vectors and 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 V^*} has the same additive structure 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} 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} is a vector space, the first four axioms will still be true. For the remaining four properties we simply check that they will hold.

5) Failed to parse (MathML with SVG or PNG fallback (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 * x = \bar{1} x = 1x = x}
6) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha * (\beta * x) = \alpha * (\bar{\beta} x) = \bar{\alpha} \bar{ \beta} x = \overline{ \alpha \beta} x = (\alpha \beta) * x}
7) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha * (x + y) = \bar{\alpha} (x+y) = \bar{\alpha} x + \bar{\alpha} y = \alpha * x + \alpha * y}
8)
Therefore Failed to parse (MathML with SVG or PNG fallback (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 complex vector space.

7. 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 P_n} be the set of polynomials 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}[t]} of degree Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq n} .
(a) 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 P_n} is a vector space.

Solution:
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 x,y \in P_n} 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 \alpha \in \mathbb{R}} . Then 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 y = b_n t^n + \cdots + b_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 x+y = (a_n+b_n)t^n + (a_0 + b_0) \in P_n} . Also Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha x = (\alpha a_n)t^n + \cdots \alpha a_0 \in P_n} 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 P_n} is closed under addition and scalar multiplication. Also Failed to parse (MathML with SVG or PNG fallback (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 = (a_n+b_n)t^n + (a_0 + b_0) = (b_n+a_n)t^n + (b_0 + a_0) = y + x} so addition is commutative. Similarly for associative. The zero 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 P_n} is the zero polynomial with all coefficients equal to 0. The additive inverse 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 x} 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 (-x) = (-a_n)t^n + (-a_0)} . Also, Failed to parse (MathML with SVG or PNG fallback (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) x = x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha(\beta x) = (\alpha \beta) x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\alpha + \beta) x = \alpha x + \beta x} 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 \alpha (x + y) = \alpha x + \alpha y} just by writing each of them out. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_n} is a vector space.


(b) Show that the space of polynomials of degree Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq 1} 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 P_n - P_{n-1}} and does not form a subspace.

Solution:
First off, the space 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 P_n - P_{n-1}} because the polynomials that have exactly degree Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} are 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 P_n} but 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 P_{n-1}} and there are no other polynomials 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 P_n} that aren’t also 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 P_{n-1}} . Now this set does not form a subspace because it is not closed under addition. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(t) = t^n} 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 q(t) = -t^n + 1} are both polynomials of degree exactly Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . However, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p+q = 1} is a polynomial of degree 0 not Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .


(c) 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 f(t): \mathbb{F} \to \mathbb{F}} , 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 = \{ p(t)f(t): p \in P_n\}} 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 Func(\mathbb{F}, \mathbb{F})} .

Solution:
To show it is a subspace, we only need to check that it is closed under addition and scalar multiplication. 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 x,y \in V} 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 \alpha \in \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 x = p(t)f(t)} for some polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in P_n} 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 y = q(t)f(t)} for some polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q \in P_n} . 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 = p(t)f(t)+q(t)f(t) = (p(t)+q(t))f(t)} . But 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 p(t)+q(t)} is just another polynomial 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 P_n} , 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} is exactly of the form polynomial times Failed to parse (MathML with SVG or PNG fallback (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(t)} . 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 x+y \in V} . Also Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha x = \alpha (p(t)f(t)) = (\alpha p(t)) f(t)} which is again of the form polynomial times Failed to parse (MathML with SVG or PNG fallback (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(t)} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha x \in V} . Therefore Failed to parse (MathML with SVG or PNG fallback (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. 8) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\alpha + \beta) * x = \overline{\alpha + \beta} x + (\bar{\alpha} + \bar{\beta}) x = \bar{\alpha}x + \bar{\beta} x = \alpha * x + \beta * x}
Therefore Failed to parse (MathML with SVG or PNG fallback (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 complex vector space.


8. 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{C}^\times = \mathbb{C} - \{0\}} . Define addition on Failed to parse (MathML with SVG or PNG fallback (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 Failed to parse (MathML with SVG or PNG fallback (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 \boxplus y = xy} . Define scalar multiplication by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \boxdot x = e^{\alpha}x} .
(a) Show that if we use Failed to parse (MathML with SVG or PNG fallback (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_V = 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 -x = x^{-1}} , then the first four axioms for a vector space are satisfied.

Solution:
1) Failed to parse (MathML with SVG or PNG fallback (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\boxplus y = xy = yx = y \boxplus x} so addition is commutative.

2) Failed to parse (MathML with SVG or PNG fallback (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 \boxplus y) \boxplus z = (xy) \boxplus z = (xy)z = xyz = x(yz) = x(y \boxplus z) = x\boxplus (y \boxplus z)} and addition is associative.
3) Failed to parse (MathML with SVG or PNG fallback (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 \boxplus 0_V = x \boxplus 1 = 1x = x}
4) Failed to parse (MathML with SVG or PNG fallback (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 \boxplus -x = x \boxplus x^{-1} = xx^{-1} = 1 = 0_V}


(b) Which of the scalar multiplication properties do not hold?

Solution:
The only property that won’t hold is associativity of scalar multiplication.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \boxdot (\beta \boxdot x) = \alpha (e^\beta x) = e^\alpha e^\beta x} which is not the same as .

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