Topics in Algebra, Chapter 4.1

2013-09-23 math algebra topics-in-algebra

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This page covers section 4.1 (“Elementary Basic Concepts” [of vector spaces and modules]).

Topics covered: 4.1

• Definition: Let $V$ be a non-empty set, let $F$ be a field, and let $+:V\times V\to V$ and $\cdot :F\times V\to V$ be binary operations such that

  1. $\alpha \cdot (v+w)=\alpha \cdot v+\alpha \cdot w$ for all $\alpha \in F$, $v,w\in V$.
  2. $(\alpha +\beta )\cdot v=\alpha \cdot v+\beta \cdot v$ for all $\alpha ,\beta \in F$, $v\in V$.
  3. $\alpha (\beta \cdot v)=(\alpha \beta )\cdot v$ for all $\alpha ,\beta \in F$, $v\in V$.
  4. $1\cdot v=v$ for all $v\in V$ where $1$ is the multiplicative unit in $F$.

  Then $V$ is said to be a vector space over $F$. The dot for multiplication will generally be omitted in what follows.

• Example: If $F\subset K$ are both fields, then $K$ may be viewed as a vector space over $F$.

• Example: If $F$ is a field, then $F^n=\{({\alpha }_1,\dots ,{\alpha }_n)\mid {\alpha }_i\in F\}$, with the obvious operations, is a vector space over $F$.

• Example: If $F$ is a field, then $F[x]$ is a vector space over $F$.

• Example: If $F$ is a field, then $P_n(F)\subset F[x]$ is a vector space over $F$, where $P_n(F)$ is the set of polynomials over $F$ with degree less than $n$.

• Definition: If $V$ is a vector space over $F$ and $W\subset V$ forms a vector space using the same operations of $V$, then $W$ is a subspace of $V$. This is equivalent to the condition that $\alpha w+{\alpha }^{\mathrm{\prime }}{w}^{\mathrm{\prime }}\in W$ for all $w,w^{\mathrm{\prime }}\in W$ and $\alpha ,{\alpha }^{\mathrm{\prime }}\in F$.

• Definition: Let $V,W$ be vector spaces over $F$. A homomorphism of vector spaces is a map $\varphi :W\to V$ such that $\varphi (v+v^{\mathrm{\prime }})=\varphi (v)+\varphi (v^{\mathrm{\prime }})$ and $\varphi (\alpha w)=\alpha \varphi (w)$ for all $w,w^{\mathrm{\prime }}\in W$ and $\alpha \in F$.

• The set of all homomorphisms between vector spaces $V$ and $W$ will be denoted $\mathrm{H}\mathrm{o}\mathrm{m}(V,W)$.

• Lemma 4.1.1: Let $V$ be a vector space over $F$. Then

  1. $\alpha 0_V=0_V$ for all $\alpha \in F$.
  2. $0_{F}v=0_V$ for all $v\in V$.
  3. $(-\alpha )v=-(\alpha v)$ for all $\alpha \in F$, $v\in V$.
  4. $\alpha v=0$ implies $\alpha =0_F$ or $v=0_V$.

• Lemma 4.1.2: Let $V$ be a vector space over $F$ and let $W\subset V$ be a subspace. Then $V\mathrm{/}W=\{v+W\mid v\in V\}$ is a vector space over $F$, called the quotient space of $V$ by $W$.

• Theorem 4.1.1: Let $V,W$ be vector spaces and let $\varphi :V\to W$ be a surjective homomorphism with kernel $K$. Then $W\cong V\mathrm{/}K$. Conversely, if $V$ is a vector space and $W\subset V$ a subspace, then there exists a homomorphism $\psi :V\to V\mathrm{/}W$. (TODO: am I transcribing this correctly?)

• Definition: Let $V$ be a vector space over $F$ and let $W_1,\dots ,W_n\subset V$ be subspaces. If any $v\in V$ admits a unique representation $v=w_1+\cdots +w_n$ with $w_i\in W_i$ for each $i$, then $V$ is the internal direct sum of the $\{W_i\}$.

• Definition: Let $V_1,\dots ,V_n$ be vector spaces over $F$. The external direct sum of the $\{V_i\}$ is the set $\{(v_1,\dots ,v_n)\mid v_i\in V_i\}$.

• Theorem 4.1.2: The internal and external direct sums of $\{V_1,\dots ,V_n\}$ are isomorphic. Hence we can refer to simply a direct sum having both of the above properties.

The problems below are paraphrased from/inspired by those given in Topics in Algebra by Herstein. The solutions are my own unless otherwise noted. I will generally try, in my solutions, to stick to the development in the text. This means that problems will not be solved using ideas and theorems presented further on in the book.

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Herstein 4.1.1

Let $V$ be a vector space over field $F$. Further, let $\alpha \in F$ and $v,w\in V$. Show that $\alpha (v-w)=\alpha v-\alpha w$ in $V$

We have $$\alpha (v-w)=\alpha (v+(-w))=\alpha v+\alpha (-w)=\alpha v-\alpha w$$

by Lemma 4.1.1.

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Herstein 4.1.2

Let $F$ be a field and $n$ a positive integer, let $V=F^n$ and let $W\subset F[x]$ be the vector space of polynomials over $F$ of degree less than $n$. Prove that $V\cong W$.

The map $\varphi :V\to W$ given by $\varphi (({\alpha }_0,\dots ,{\alpha }_{n-1}))={\alpha }_{n-1}{x}^{n-1}+\dots +{\alpha }_0$ is an isomorphism.

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Herstein 4.1.3

Prove that the kernel of homomorphism is a subspace.

For $\varphi :V\to W$ a homomorphism between vector spaces $V,W$, $\ker \varphi =\{v\in W\mid \varphi (v)=0\}$. Let $v,v^{\mathrm{\prime }}\in \ker \varphi$ and $\alpha \in F$, with $F$ the base field. We have $\varphi (v+\alpha v^{\mathrm{\prime }})=\varphi (v)+\alpha \varphi (v^{\mathrm{\prime }})$ because $\varphi$ is a homomorphism, and so $\varphi (v+\alpha v^{\mathrm{\prime }})=0$ and $v+\alpha v^{\mathrm{\prime }}\in \ker \varphi$. Thus $\ker \varphi$ is a subspace of $V$.

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Herstein 4.1.4

(a) Show that the set $V$ of continuous functions $[0,1]\to \mathbb{R}$ is a vector space over $\mathbb{R}$.

(b) For positive integer $n$, show that the set of functions $[0,1]\to \mathbb{R}$, for which the first $n$ derivatives exist, form a subspace of the vector space from (a).

(a) Let $f,g\in V$ and $\alpha \in \mathbb{R}$. We have $f+\alpha g\in V$ because sums and scalar products of continuous functions are again continuous. The function $0:x\mapsto 0$ is the additive identity in this vector space. Other details can be taken for granted.

(b) The set of $n$-times differentiable functions is a subset of the continuous functions, so it’s just necessary to check if the set is closed under linear combinations. Indeed, the sum of a differentiable function is again differentiable, so the set in question is a subspace of $V$.

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Herstein 4.1.5

(a) Let $V=\{(a_1,a_2,\dots )\mid a_i\in \mathbb{R}\}$ with all operations defined componentwise. Show that $V$ is a vector space over $\mathbb{R}$.

(b) Let $W=\{(a_1,\dots ,a_n,\dots )\in V\mid {\lim }_{n\to \mathrm{\infty }}{a}_n=0\}$. Prove that $W$ is a subspace of $V$.

(c)* Let $U=\{(a_1,\dots ,a_n,\dots )\in V\mid {\sum }_{i=1}^{\mathrm{\infty }}{a}_i^2<\mathrm{\infty }\}$. Prove that $U$ is a subspace of $V$ and that $U$ is contained in $W$.

(a) Because $\mathbb{R}$ is closed under addition and multiplication, and operations on $V$ are defined componentwise, all vector space axioms hold for $V$.

(b) If $(a_i)$ and $(b_i)$ are two elements of $W$ and $\alpha \in \mathbb{R}$, then $(a_i+\alpha b_i)\in W$ because $$\underset{i\to \mathrm{\infty }}{\lim }(a_i+\alpha b_i)=\underset{i\to \mathrm{\infty }}{\lim }{a}_i+\alpha \underset{i\to \mathrm{\infty }}{\lim }{b}_i=0.$$

(c) Let $(a_i),(b_i)\in U$ and let $\alpha \in \mathbb{R}$. We have that $$\sum _i(a_i+\alpha b_i{)}^2=\sum _i{a}_i^2+{\alpha }^2\sum _i{b}_i^2+2\alpha \sum _i{a}_i{b}_i\mathrm{.}$$

The first two terms are finite by assumption. The third term can be bounded: for real numbers $x,y$, rearrange $(x-y{)}^2\ge 0$ to give $$xy\le \frac{1}{2}{x}^2+\frac{1}{2}{y}^2$$

so that $$\sum _i{a}_i{b}_i\le \frac{1}{2}\sum _i{a}_i^2+\frac{1}{2}\sum _i{b}_i^2<\mathrm{\infty }\mathrm{.}$$

Hence $(a_i+\alpha b_i)\in U$ and $U$ is a subspace of $V$.

To show that $U$ is contained in $W$, we must show that ${\sum }_{i=1}^{\mathrm{\infty }}{a}_i^2<\mathrm{\infty }$ implies ${\lim }_{i\to \mathrm{\infty }}{a}_i=0$. Define the partial sums $s_n={\sum }_{i=1}^n{a}_i^2$; we have that ${\lim }_{n\to \mathrm{\infty }}{s}_n={\lim }_{n\to \mathrm{\infty }}{s}_{n-1}=L$ for some $L<\mathrm{\infty }$. Therefore, $$0=\underset{n\to \mathrm{\infty }}{\lim }(s_n-s_{n-1})=\underset{n\to \mathrm{\infty }}{\lim }{a}_n^2\mathrm{.}$$

For any $ϵ>0$, there exists $N\in \mathbb{N}$ such that $n>N$ implies that $a_n^2<ϵ$. Let ${ϵ}^{\mathrm{\prime }}>0$ be given: by the previous statement, there exists $N\in \mathbb{N}$ so that $n>N$ implies $a_n^2<{ϵ}^{\mathrm{\prime }2}$. Thus for $n>N$, we also have $\mathrm{\mid }{a}_n\mathrm{\mid }<{ϵ}^{\mathrm{\prime }}$. This proves that ${\lim }_{n\to \mathrm{\infty }}{a}_n=0$, i.e. that $(a_i)\in W$.

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Herstein 4.1.6

Let $U,V$ be vector spaces over field $F$. Define operations on $\mathrm{H}\mathrm{o}\mathrm{m}(U,V)$ to make it into a vector space over $F$.

$\mathrm{H}\mathrm{o}\mathrm{m}(U,V)$ is the set of homomorphisms $U\to V$. Given $\varphi ,\psi \in \mathrm{H}\mathrm{o}\mathrm{m}(U,V)$ and $\alpha \in F$, we can define a third homomorphism pointwise, i.e. by $$(\varphi +\alpha \psi )(u)=\varphi (u)+\alpha \psi (u)\mathrm{.}$$

It is straightforward to see that $\varphi +\alpha \psi$ is again a homomorphism. $\mathrm{H}\mathrm{o}\mathrm{m}(U,V)$ is a vector space under this pointwise addition and scalar multiplication.

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Herstein 4.1.7*

With $F$ a field, prove that $\mathrm{H}\mathrm{o}\mathrm{m}(F^n,F^m)$ is isomorphic to $F^{mn}$ as vector spaces.

Let $e_i^{(n)}\in F^n$ be the vector with a $1$ in the $i$-th index and zeroes elsewhere. Similarly, let $e_i^{(m)}\in F^m$ be the analogous thing. Given $\varphi \in \mathrm{H}\mathrm{o}\mathrm{m}(F^n,F^m)$, we have $\varphi (e_i^{(n)})={\sum }_j{\alpha }_{ij}^{(\varphi )}{e}_j^{(m)}$, defining a matrix of coefficients ${\alpha }_{ij}^{(\varphi )}\in F$ for each $\varphi$. Now define $f:\mathrm{H}\mathrm{o}\mathrm{m}(F^n,F^m)\to F^{mn}$ by $$f(\varphi )=({\alpha }_{11}^{(\varphi )},\dots ,{\alpha }_{1m}^{(\varphi )},{\alpha }_{21}^{(\varphi )},\dots ,{\alpha }_{nm}^{(\varphi )})\mathrm{.}$$

That $f$ respects linear combinations is a rote computation. The kernel of $f$ is trivial, so it is an isomorphism.

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Herstein 4.1.8

Let $F$ be a field and $n>m$ be positive integers. Exhibit a surjective homomorphism $F^n\to F^m$ and show that its kernel is isomorphic to $F^{n-m}$.

Define $\varphi :F^n\to F^m$ by $(a_1,\dots ,a_m,a_{m+1},\dots ,a_n)\mapsto (a_1,\dots ,a_m)$. This is a surjective homomorphism. The kernel of $\varphi$ is the set $\{(0,\dots ,0,a_{m+1},\dots ,a_n)\in F^n\}$; a similar projection mapping establishes the isomorphism $\ker \varphi \cong F^{n-m}$.

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Herstein 4.1.9

Fix nonzero $v\in F^n$. Show there exists $\varphi \in \mathrm{H}\mathrm{o}\mathrm{m}(F^n,F)$ with $\varphi (v)\mathrm{\ne }0$.

Let $m$ be the index of the first non-zero entry in $v$, and let $\varphi$ be the projection of $F^n$ onto its $m$-th entry. Then $\varphi \in \mathrm{H}\mathrm{o}\mathrm{m}(F^n,F)$ and $\varphi (v)\mathrm{\ne }0$.

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Herstein 4.1.10

With $F$ a field and $n$ a positive integer, prove that $F^n\cong \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{H}\mathrm{o}\mathrm{m}(F^n,F),F)$.

This is (a special case of) the result that a vector space is isomorphic to its double-dual.

With the result 4.1.7, we have $$\mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{H}\mathrm{o}\mathrm{m}(F^n,F),F)\cong \mathrm{H}\mathrm{o}\mathrm{m}(F^n,F)\cong F^n\mathrm{.}$$

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Herstein 4.1.11

With $U,W$ subspaces of vector space $V$, all over field $F$, prove that $U+W=\{u+w\mid u\in U,w\in W\}$ is a subspace of $V$.

Given $u,u^{\mathrm{\prime }}\in U$, $w,w^{\mathrm{\prime }}\in W$ and $\alpha \in F$, we have that $$(u+w)+\alpha (u^{\mathrm{\prime }}+w^{\mathrm{\prime }})=(u+\alpha u^{\mathrm{\prime }})+(w+\alpha w^{\mathrm{\prime }})=u^{\mathrm{\prime }\mathrm{\prime }}+w^{\mathrm{\prime }\mathrm{\prime }}\in U+W,$$

where the last step is justified because $U$ and $W$ are each subspaces. Therefore $U+W$ is a subspace of $V$.

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Herstein 4.1.12

Prove that the intersection of two subspaces of $V$ is again a subspace of $V$.

Let $U,W$ be subspaces of $V$ over the field $F$. If $v,v^{\mathrm{\prime }}\in U\cap W$ and $\alpha \in F$, then $v+\alpha v^{\mathrm{\prime }}\in U$ because $U$ is a subspace and $v+\alpha v^{\mathrm{\prime }}\in W$ because $W$ is a subspace. Hence $v+\alpha v^{\mathrm{\prime }}\in U\cap W$, and $U\cap W$ is a subspace.

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Herstein 4.1.13

With $U,W$ subspaces of vector space $V$, all over field $F$, prove that $(U+W)\mathrm{/}W\cong U\mathrm{/}(U\cap W)$.

This is the second isomorphism theorem.

The elements of $(U+W)\mathrm{/}W$ look like $u+w+W=u+W$ where $u\in U$ and $w\in W$. The elements of $U\mathrm{/}(U\cap W)$ look like $u+U\cap W$ with $u\in U$. In both cases, elements of $U\cap W$ get turned into the zero coset.

Define the map $\varphi :(U+W)\mathrm{/}W\to U\mathrm{/}(U\cap W)$ by $\varphi (u+W)=u+U\cap W$. To see that this is well-defined, consider $u+w,u^{\mathrm{\prime }}+w^{\mathrm{\prime }}\in U+W$ that belong to the same coset: $u+w+W=u^{\mathrm{\prime }}+w^{\mathrm{\prime }}+W$, so that their difference is $u-u^{\mathrm{\prime }}\in W$ which then implies that $u-u^{\mathrm{\prime }}\in U\cap W$. We have $\varphi (u+w+W)-\varphi (u^{\mathrm{\prime }}+w^{\mathrm{\prime }}+W)=(u-u^{\mathrm{\prime }})+U\cap W=U\cap W$; thus any representative of a coset in the domain gets mapped to the same coset in the codomain.

$\varphi$ is a homomorphism: for $u,u^{\mathrm{\prime }}\in U$, $w,w^{\mathrm{\prime }}\in W$, we have $\varphi (u+w+\alpha (u^{\mathrm{\prime }}+w^{\mathrm{\prime }})+W)=u+\alpha u^{\mathrm{\prime }}+U\cap W$ while $\varphi (u+w+W)+\alpha \varphi (u^{\mathrm{\prime }}+w^{\mathrm{\prime }}+W)=(u+U\cap W)+\alpha (u^{\mathrm{\prime }}+U\cap W)=u+\alpha u^{\mathrm{\prime }}+U\cap W$.

The kernel of $\varphi$ contains those elements which map to $0$ in the codomain, i.e. those elements of the domain where the $U$ component belongs to $U\cap W$. We have that $\{u+w\mid u\in U\cap W,w\in W\}\subset W$ so $\ker \varphi =W$, i.e. the kernel is trivial so that $\varphi$ is injective.

$\varphi$ is surjective because, given $u+U\cap W\in U\mathrm{/}(U\cap W)$, we have $\varphi (u+W)=u+U\cap W$.

Therefore $\varphi :(U+W)\mathrm{/}W\to U\mathrm{/}(U\cap W)$ is an isomorphism.

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Herstein 4.1.14

Let $U,V$ be vector spaces and let $\varphi :U\to V$ be a surjective homomorphism. Show there is a one-to-one correspondence between $\mathcal{A}$, the subspaces of $V$, and $\mathcal{B}$, the subspaces of $U$ which contain $\ker \varphi$.

This is the fourth (“lattice”) isomorphism theorem.

There are a couple of natural-looking ways to map the objects in question (I tried $W\mapsto \varphi (W)$, $W\mapsto U\mathrm{/}W$, etc.). However, the first isomorphism theorem (theorem 4.1.1) states that $V\cong U\mathrm{/}\ker \varphi$, so the subspaces of $V$ should probably look like $W\mathrm{/}\ker \varphi$ where $W$ is a subspace of $V$. Naturally, $W\mathrm{/}\ker \varphi$ only makes sense if $W$ contains $\ker \varphi$. Therefore, the map we define is $f:\mathcal{B}\to \mathcal{A}$ given by $f(W)=W\mathrm{/}\ker \varphi$, and it makes sense because of the way we have chosen $\mathcal{B}$ (i.e. only considering subspaces that contain $\ker \varphi$).

The map $f$ is injective: let $W,W^{\mathrm{\prime }}\in \mathcal{B}$ be mapped the same by $f$, i.e. $W\mathrm{/}\ker \varphi =W^{\mathrm{\prime }}\mathrm{/}\ker \varphi$. We would like to show that this implies $W=W^{\mathrm{\prime }}$. If $w\in W$, then $w+\ker \varphi \in W\mathrm{/}\ker \varphi =W^{\mathrm{\prime }}\mathrm{/}\ker \varphi$ so there exists $w^{\mathrm{\prime }}\in W^{\mathrm{\prime }}$ with $w+\ker \varphi =w^{\mathrm{\prime }}+\ker \varphi$. This implies that $w-w^{\mathrm{\prime }}\in \ker \varphi \subset W^{\mathrm{\prime }}$, so that $$w=(w-w^{\mathrm{\prime }})+w^{\mathrm{\prime }}\in W^{\mathrm{\prime }}\mathrm{.}$$

This proves that $W\subset W^{\mathrm{\prime }}$. The argument, made in reverse, gives also that $W^{\mathrm{\prime }}\subset W$, so we have proven that $f$ is injective.

$f$ is also surjective: if $X$ is a subspace of $V$, then we can realize it as the image of a subspace of $U$. Consider $Y={\varphi }^{-1}(X)=\{u\in U\mid \varphi (u)\in X\}$. It remains to show that $\ker \varphi \subset Y$, that $Y$ is a subspace of $U$, and that $f(Y)=X$. Because $X$ is a subspace, $0\in X$ and $\varphi (\ker \varphi )=\{0\}\subset X$, so that $\ker \varphi \subset Y$. If $y,y^{\mathrm{\prime }}\in Y$ and $\alpha$ is a scalar, then $\varphi (y+\alpha y^{\mathrm{\prime }})=\varphi (y)+\alpha \varphi (y^{\mathrm{\prime }})\in X$, so $Y$ is a subspace of $U$. Finally, $f(Y)=\{u+\ker \varphi \mid u\in U,\varphi (u)\in X\}$, which is the same thing as $X$ (TODO: fill in details without making notation worse?).

Therefore, $f$ is a bijection between $\mathcal{A}$ and $\mathcal{B}$.

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Herstein 4.1.15

Let $V$ be a vector space and let $V_1,\dots ,V_n$ be subspaces of $V$ such that $V=V_1+\cdots +V_n$ and $V_i\cap {\sum }_{j\mathrm{\ne }i}{V}_j=\{0\}$ for every $i$. Prove that $V$ is the internal direct sum of $V_1,\dots ,V_n$.

To say that $V$ is the internal direct sum of the $V_i$ is to say that $v\in V$ has exactly one expression $v=v_1+\cdots +v_n$ with each $v_i\in V_i$.

Because $V\subset {\sum }_i{V}_i$, we have that $v\in V$ has at least one such expression. It remains to show that this expression is unique. Therefore, suppose that $v=v_1+\cdots +v_n=v_1^{\mathrm{\prime }}+\cdots +v_n^{\mathrm{\prime }}$ with $v_i,v_i^{\mathrm{\prime }}\in V_i$ for each $i$. Then we have $$(v_1-v_1^{\mathrm{\prime }})+\cdots +(v_n-v_n^{\mathrm{\prime }})=0,$$

and, rearranging, $$v_i-v_i^{\mathrm{\prime }}=-\sum _{j\mathrm{\ne }i}(v_j-v_j^{\mathrm{\prime }})\mathrm{.}$$

The left hand side belongs to $V_i$ while the right hand side belongs to ${\sum }_{j\mathrm{\ne }i}{V}_j$. By assumption, those two spaces intersect trivially, so that $v_i-v_i^{\mathrm{\prime }}=0$ for each $i$. Hence the two representations are identical, and we are done.

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Herstein 4.1.16

Let $V$ be a vector space with subspaces $V_1,\dots ,V_n$ such that $V={\oplus }_i{V}_i$. Prove there are subspaces ${\bar{V}}_i\subset V$ isomorphic to $V_i$ with $V$ the internal direct sum of ${\bar{V}}_i$.

$V$ is the external direct sum of the $V_i$, so it looks like $$V=\{(v_1,\dots ,v_n)\mid v_i\in V_i\}\mathrm{.}$$

The subspaces ${\bar{V}}_i$ which allow the $i$-th entry to range over $V_i$, while fixing the non-$i$ entries as zero, are the desired subspaces of $V$ isomorphic to $V_i$. The conditions of exercise 4.1.15 are easily satisfied, so that $V$ is the internal direct sum of the ${\bar{V}}_i$.

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Herstein 4.1.17

Let $F$ be a field and let $T:F^2\to F^2$ be defined by $T(x_1,x_2)=(\alpha x_1+\beta x_2,\gamma x_1+\delta x_2)$ for fixed $\alpha ,\beta ,\gamma ,\delta \in F$.

(a) Prove that $T$ is a homomorphism.

(b) Find necessary and sufficient conditions on $\alpha ,\beta ,\gamma ,\delta$ so that $T$ is an isomorphism.

(a) That $T$ is a homomorphism is a straightforward exercise

(b) In the language of matrices, this is the familiar question of when a matrix is invertible; the answer is “when the determinant is non-zero”. How does that come about from direct computation?

Let $y_1,y_2\in F$ and consider the simultaneous equations $$\alpha x_1+\beta x_2=y_1,$$

$$\gamma x_1+\delta x_2=y_2\mathrm{.}$$

Multiplying the first equation by $\delta$ and the second by $\beta$, and then subtracting the second from the first, we find $$(\alpha \delta -\beta \gamma )x_1=\delta y_1-\beta y_2\mathrm{.}$$

Performing a similar computation, we also find $$(\alpha \delta -\beta \gamma )x_2=\alpha y_2-\gamma y_1\mathrm{.}$$

In order for $T$ to be injective, it must have a trivial kernel. If $(x_1,x_2)\in \ker T$, then $$(\alpha \delta -\beta \gamma )x_1=(\alpha \delta -\beta \gamma )x_2=0.$$

These equations have non-trivial solutions $(x_1,x_2)$ if and only if $\alpha \delta -\beta \gamma =0$. Thus a necessary and sufficient condition for $T$ to be injective is that $\alpha \delta -\beta \gamma \mathrm{\ne }0$. As a result, this is also a necessary condition for $T$ to be an isomorphism.

The same condition is also sufficient for $T$ to be surjective, because the equations $(\alpha \delta -\beta \gamma )x_1=\delta y_1-\beta y_2$ and $(\alpha \delta -\beta \gamma )x_2=\alpha y_2-\gamma y_1$ are solvable for any $y_1,y_2$ by dividing by $\alpha \delta -\beta \gamma$.

Therefore the necessary and sufficient condition for $T$ to be an isomorphism is that $\alpha \delta -\beta \gamma$ be non-zero.

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Herstein 4.1.18

The same exercise as 4.1.17 but on $F^3$.

I haven’t done this exercise, but I would be surprised if it is different from 4.1.17 in a meaningful way.

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Herstein 4.1.19

Let $V,W$ be vector spaces and let $T:V\to W$. Use $T$ to define a homomorphism $T^{\ast }:\mathrm{H}\mathrm{o}\mathrm{m}(W,F)\to \mathrm{H}\mathrm{o}\mathrm{m}(V,F)$.

Put another way, the exercise is to show that a homomorphism between vector spaces $V$ and $W$ induces a natural homomorphism between their dual spaces $V^{\ast }=\mathrm{H}\mathrm{o}\mathrm{m}(V,F)$ and $W^{\ast }=\mathrm{H}\mathrm{o}\mathrm{m}(W,F)$.

A diagram helps:

$$\begin{array}{ccc} V & \rightarrow & F \ & \searrow & \uparrow \ & & W \end{array}$$

Here, the map $V\to W$ is provided by $T$, the map $W\to F$ is some representative $w^{\ast }\in \mathrm{H}\mathrm{o}\mathrm{m}(W,F)$, and the desired map $V\to F$ can be made in a natural way by composition. That is, we define a map $T^{\ast }:\mathrm{H}\mathrm{o}\mathrm{m}(W,F)\to \mathrm{H}\mathrm{o}\mathrm{m}(V,F)$ by $$T^{\ast }(w^{\ast })=w^{\ast }\circ T\mathrm{.}$$

It is easy to check that (1) the resulting $v^{\ast }=w^{\ast }\circ T$ is indeed an element of $\mathrm{H}\mathrm{o}\mathrm{m}(V,F)$, and (2) that $T^{\ast }$ is a homomorphism.

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Herstein 4.1.20

Let $F$ be a field.

(a) Prove that $F$ is not isomorphic to $F^n$ for integer $n>1$.

(b) Prove that $F^2$ is not isomorphic to $F^3$.

(a) Looking slightly ahead, the intuition here is that the image of $F$ under a homomorphism will be too low-dimensional. Therefore, consider a supposed isomorphism $\varphi :F\to F^n$. Because it is surjective, there are $f_1,f_2\in F$ with $\varphi (f_1)=(1,0,\dots ,0)$ and $\varphi (f_2)=(0,1,0,\dots ,0)$. Now, there exists $\alpha \in F$ such that $f_2=\alpha f_1$, so we must have $$(0,1,0,\dots ,0)=\varphi (f_2)=\varphi (\alpha f_1)=\alpha \varphi (f_1)=(\alpha ,0,\dots ,0)\mathrm{.}$$

This is a contradiction, so we conclude that no such $\varphi$ exists.

(b) Suppose $\varphi :F^2\to F^3$ is an isomorphism, $\varphi ((1,0))=v_1\in F^3$ and $\varphi ((0,1))=v_2\in F^3$. Then we have $\varphi ((\alpha ,\beta ))=\alpha v_1+\beta v_2$ for any $\alpha ,\beta \in F$. Because $\varphi$ is surjective, there must exist ${\alpha }_i,{\beta }_i$ such that $${\alpha }_1{v}_1+{\beta }_1{v}_2=(1,0,0),$$

$${\alpha }_2{v}_1+{\beta }_2{v}_2=(0,1,0),$$

$${\alpha }_3{v}_1+{\beta }_3{v}_2=(0,0,1)\mathrm{.}$$

Taking the first and second equations, and eliminating the $v_2$ terms, we find that $$({\alpha }_1{\beta }_2-{\beta }_1{\alpha }_2)v_1=({\beta }_2,-{\beta }_1,0)\mathrm{.}$$

However, taking the first and third equations, and eliminating the $v_2$ terms, we also find that $$({\alpha }_1{\beta }_3-{\beta }_1{\alpha }_3)v_1=({\beta }_3,0,-{\beta }_1)\mathrm{.}$$

These two results are inconsistent unless ${\beta }_1=0$. In that case, we can explicitly solve for $v_1=({\alpha }_1^{-1},0,0)$ and derive a contradiction that ${\beta }_2{v}_2=(-\frac{{\alpha }_2}{{\alpha }_1},1,0)$ while ${\beta }_3{v}_2=(-\frac{{\alpha }_3}{{\alpha }_1},0,1)$.

Thus we have shown that the map $\varphi$ is not truly surjective, and therefore not an isomorphism.

The laborious arguments above make one appreciate (1) the elegance of doing linear algebra without explicit coordinates/choice of basis, and (2) the simplicity and utility of the concepts of linear independence, basis and dimension, which we eschew here because they are not introduced until the next section of the book.

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Herstein 4.1.21

Let $V$ be a vector space over the infinite field $F$. Prove that $V$ is not realizable as the set-theoretic union of a finite number of its proper subspaces.

Let $V_1,\dots ,V_n$ be proper subspaces of $V$ such that ${\bigcup }_i{V}_i=V$. We can assume that each $V_i$ brings something of value to this union, i.e. that $$V_i\subset ̸\bigcup _{j\mathrm{\ne }i}{V}_j$$

In other words, for each $i$, there exists some $v_i$ which only belongs to $V_i$ and none of the other subspaces. If this is not the case, then we can omit this $V_i$: all of its elements are included elsewhere. In this sense, we can assume our set to have minimal size.

Because the subspaces are proper, we know that $n\ge 2$. Consider elements $v_1\in V_1$ with $v_1\in ̸{\bigcup }_{i\mathrm{\ne }1}{V}_i$ and $v_2\in V_2$ with $v_2\in ̸{\bigcup }_{i\mathrm{\ne }2}{V}_i$. Let $\alpha ,\beta \in F$ be distinct. The elements $x=v_1+\alpha v_2$ and $y=v_1+\beta v_2$ belong to $V={\bigcup }_i{V}_i$, so each belongs to some $V_i$. Suppose $x,y$ both belong to the same $V_i$; then so must their difference: $x-y=(\alpha -\beta )v_2\in V_i$. By assumption, $v_2$ only belongs to $V_2$, so that $V_i=V_2$. Taking a step back, we see that this would force $v_1=x-\alpha v_2$ to also live in $V_2$, a contradiction. Thus $x$ and $y$ are forced to belong to different subspaces.

Now, we enumerate some infinite subset $\{{\alpha }_1,{\alpha }_2,\dots \}$ of $F$ and construct the elements $x_i=v_1+{\alpha }_i{v}_2\in V$. Considering the $\{x_i\}$ pairwise, we see that every one must live in a different subspace from every other one: no finite number of subspaces will suffice. We conclude that no vector space over an infinite field can be realized as the union of finitely many of its proper subspaces.