Definition and Examples

Verify, using Example 1.4 as a model, that the two correspondences given before the definition are isomorphisms.

• (a)  Example 1.1     (b)  Example 1.2

For the map $f\colon\mathcal{P}_{1}\to\mathbb{R}^{2}$ given by

Find the image of each of these elements of the domain.

• (a)  $3-2x$     (b)  $2+2x$     (c)  $x$

Show that this map is an isomorphism.

Show that the natural map $f_{1}$ from Example 1.5 is an isomorphism.

Show that the map $t\colon\mathcal{P}_{2}\to\mathcal{P}_{2}$ given by $t(ax^{2}+bx+c)=bx^{2}-(a+c)x+a$ is an isomorphism.

Verify that this map is an isomorphism: $h\colon\mathbb{R}^{4}\to\mathcal{M}_{2\!\times\!2}$ given by

Decide whether each map is an isomorphism. If it is an isomorphism then prove it and if it isn’t then state a condition that it fails to satisfy.

• $f\colon\mathcal{M}_{2\!\times\!2}\to\mathbb{R}$ given by

• $f\colon\mathcal{M}_{2\!\times\!2}\to\mathbb{R}^{4}$ given by

• $f\colon\mathcal{M}_{2\!\times\!2}\to\mathcal{P}_{3}$ given by

• $f\colon\mathcal{M}_{2\!\times\!2}\to\mathcal{P}_{3}$ given by

Show that the map $f\colon\mathbb{R}^{1}\to\mathbb{R}^{1}$ given by $f(x)=x^{3}$ is one-to-one and onto. Is it an isomorphism?

Refer to Example 1.1. Produce two more isomorphisms (of course, you must also verify that they satisfy the conditions in the definition of isomorphism).

Refer to Example 1.2. Produce two more isomorphisms (and verify that they satisfy the conditions).

Show that, although $\mathbb{R}^{2}$ is not itself a subspace of $\mathbb{R}^{3}$, it is isomorphic to the $xy$-plane subspace of $\mathbb{R}^{3}$.

Find two isomorphisms between $\mathbb{R}^{16}$ and $\mathcal{M}_{4\!\times\!4}$.

For what $k$ is $\mathcal{M}_{m\!\times\!n}$ isomorphic to $\mathbb{R}^{k}$?

For what $k$ is $\mathcal{P}_{k}$ isomorphic to $\mathbb{R}^{n}$?

Prove that the map in Example 1.9, from $\mathcal{P}_{5}$ to $\mathcal{P}_{5}$ given by $p(x)\mapsto p(x-1)$, is a vector space isomorphism.

Why, in Lemma 1.10, must there be a $\vec{v}\in V$? That is, why must $V$ be nonempty?

Are any two trivial spaces isomorphic?

In the proof of Lemma 1.11, what about the zero-summands case (that is, if $n$ is zero)?

Show that any isomorphism $f\colon\mathcal{P}_{0}\to\mathbb{R}^{1}$ has the form $a\mapsto ka$ for some nonzero real number $k$.

These prove that isomorphism is an equivalence relation.

• Show that the identity map $\text{id}\colon V\to V$ is an isomorphism. Thus, any vector space is isomorphic to itself.

• Show that if $f\colon V\to W$ is an isomorphism then so is its inverse $f^{-1}\colon W\to V$. Thus, if $V$ is isomorphic to $W$ then also $W$ is isomorphic to $V$.

• Show that a composition of isomorphisms is an isomorphism: if $f\colon V\to W$ is an isomorphism and $g\colon W\to U$ is an isomorphism then so also i…

Suppose that $f\colon V\to W$ preserves structure. Show that $f$ is one-to-one if and only if the unique member of $V$ mapped by $f$ to $\vec{0}_{W}$ is $\vec{0}_{V}$.

Suppose that $f\colon V\to W$ is an isomorphism. Prove that the set $\text{$\{\hskip 1.00006pt\vec{v}_{1},\dots,\vec{v}_{k}\hskip 1.00006pt\}$}\subseteq V$ is linearly dependent if and only if the set of images $\text{$\{\hskip 1.00006ptf(\vec{v}_{1}),\dots,f(\vec{v}_{k})\hskip 1.00006pt\}$}\subseteq W$ is linearly dependent.

Produce an automorphism of $\mathcal{P}_{2}$ other than the identity map, and other than a shift map $p(x)\mapsto p(x-k)$.

• Show that a function $f\colon\mathbb{R}^{1}\to\mathbb{R}^{1}$ is an automorphism if and only if it has the form $x\mapsto kx$ for some $k\neq 0$.

• Let $f$ be an automorphism of $\mathbb{R}^{1}$ such that $f(3)=7$. Find $f(-2)$.

• Show that a function $f\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is an automorphism if and only if it has the form

  for some $a,b,c,d\in\mathbb{R}$ with $ad-bc\neq 0$. Hint. Exercises in prior subsections have shown that

  if and only if $ad-bc\neq 0$.

• Let $f$ be an automorphism of $\mathbb{R}^{2}$ with

  Find

Refer to Lemma 1.10 and Lemma 1.11. Find two more things preserved by isomorphism.

We show that isomorphisms can be tailored to fit in that, sometimes, given vectors in the domain and in the range we can produce an isomorphism associating those vectors.

• Let $B=\langle\vec{\beta}_{1},\vec{\beta}_{2},\vec{\beta}_{3}\rangle$ be a basis for $\mathcal{P}_{2}$ so that any $\vec{p}\in\mathcal{P}_{2}$ has a unique representation as $\vec{p}=c_{1}\vec{\beta}_{1}+c_{2}\vec{\beta}_{2}+c_{3}\vec{\beta}_{3}$, which we denote in this way.

  Show that the ${\rm Rep}_{B}(\cdot)$ operation is a function from $\mathcal{P}_{2}$ to $\mathbb{R}^{3}$ (this entails showing that with every domain vector $\vec{v}\in\mathcal{P}_{2}$ there is an associat…

Prove that a space is $n$-dimensional if and only if it is isomorphic to $\mathbb{R}^{n}$. Hint. Fix a basis $B$ for the space and consider the map sending a vector over to its representation with respect to $B$.

(Requires the subsection on Combining Subspaces, which is optional.) Let $U$ and $W$ be vector spaces. Define a new vector space, consisting of the set $U\times W=\text{$\{\hskip 1.00006pt(\vec{u},\vec{w})\mid\vec{u}\in U\text{\ and\ }\vec{w}\in W\hskip 1.00006pt\}$}$ along with these operations.

This is a vector space, the external direct sum of $U$ and $W$.

• Check that it is a vector space.

• Find a basis for, and the dimension of, the external direct sum $\mathcal{P}_{2}\times\mathbb{R}^{2}$.

• What is the relationship among $\dim(U)$, $\dim(W)$, and $\dim(U\times W)$?

• Suppose…