Section 1.4 Homework

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) Failed to parse (MathML with SVG or PNG fallback (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.

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 $\alpha \in \mathbb {R}$. Then $x=a_{n}t^{n}+\cdots +a_{0}$ and $y=b_{n}t^{n}+\cdots +b_{0}$. So that $x+y=(a_{n}+b_{n})t^{n}+(a_{0}+b_{0})\in P_{n}$. Also $\alpha x=(\alpha a_{n})t^{n}+\cdots \alpha a_{0}\in P_{n}$ so that $P_{n}$ is closed under addition and scalar multiplication. Also $x+y=(a_{n}+b_{n})t^{n}+(a_{0}+b_{0})=(b_{n}+a_{n})t^{n}+(b_{0}+a_{0})=y+x$ so addition is commutative. Similarly for associative. The zero vector in $P_{n}$ is the zero polynomial with all coefficients equal to 0. The additive inverse of $x$ is $(-x)=(-a_{n})t^{n}+(-a_{0})$. Also, $(1)x=x$, $\alpha (\beta x)=(\alpha \beta )x$, $(\alpha +\beta )x=\alpha x+\beta x$ and $\alpha (x+y)=\alpha x+\alpha y$ just by writing each of them out. Therefore $P_{n}$ is a vector space.

(b) Show that the space of polynomials of degree $n\geq 1$ is $P_{n}-P_{n-1}$ and does not form a subspace.

Solution First off, the space is equal to $P_{n}-P_{n-1}$ because the polynomials that have exactly degree $n$ are in $P_{n}$ but not in $P_{n-1}$ and there are no other polynomials in $P_{n}$ that aren’t also in $P_{n-1}$. Now this set does not form a subspace because it is not closed under addition. $p(t)=t^{n}$ and $q(t)=-t^{n}+1$ are both polynomials of degree exactly $n$. However, $p+q=1$ is a polynomial of degree 0 not $n$.

(c) If $f(t):\mathbb {F} \to \mathbb {F}$, show that $V=\{p(t)f(t):p\in P_{n}\}$ is a subspace of $Func(\mathbb {F} ,\mathbb {F} )$.

Solution To show it is a subspace, we only need to check that it is closed under addition and scalar multiplication. Let $x,y\in V$ and $\alpha \in \mathbb {F}$. Then $x=p(t)f(t)$ for some polynomial $p\in P_{n}$ and $y=q(t)f(t)$ for some polynomial $q\in P_{n}$. Then $x+y=p(t)f(t)+q(t)f(t)=(p(t)+q(t))f(t)$. But since $p(t)+q(t)$ is just another polynomial in $P_{n}$, then $x+y$ is exactly of the form polynomial times $f(t)$. Thus $x+y\in V$. Also $\alpha x=\alpha (p(t)f(t))=(\alpha p(t))f(t)$ which is again of the form polynomial times $f(t)$ so $\alpha x\in V$. Therefore $V$ is a subspace. 8) $(\alpha +\beta )*x={\overline {\alpha +\beta }}x+({\bar {\alpha }}+{\bar {\beta }})x={\bar {\alpha }}x+{\bar {\beta }}x=\alpha *x+\beta *x$
Therefore $V^{*}$ is a complex vector space.

8. Let $V=\mathbb {C} ^{\times }=\mathbb {C} -\{0\}$. Define addition on $V$ by $x\boxplus y=xy$. Define scalar multiplication by $\alpha \boxdot x=e^{\alpha }x$.
(a) Show that if we use $0_{V}=1$ and $-x=x^{-1}$, then the first four axioms for a vector space are satisfied.

Solution 1) $x\boxplus y=xy=yx=y\boxplus x$ so addition is commutative.
2) $(x\boxplus y)\boxplus z=(xy)\boxplus z=(xy)z=xyz=x(yz)=x(y\boxplus z)=x\boxplus (y\boxplus z)$ and addition is associative.
3) $x\boxplus 0_{V}=x\boxplus 1=1x=x$
4) $x\boxplus -x=x\boxplus x^{-1}=xx^{-1}=1=0_{V}$

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

Solution The only property that won’t hold is associativity of scalar multiplication.
$\alpha \boxdot (\beta \boxdot x)=\alpha (e^{\beta }x)=e^{\alpha }e^{\beta }x$ which is not the same as $(\alpha \beta )\boxdot x=e^{\alpha \beta }x$.