Dimension Characterizes Isomorphism

Decide if the spaces are isomorphic.

• (a)  $\mathbb{R}^{2}$, $\mathbb{R}^{4}$     (b)  $\mathcal{P}_{5}$, $\mathbb{R}^{5}$     (c)  $\mathcal{M}_{2\!\times\!3}$, $\mathbb{R}^{6}$     (d)  $\mathcal{P}_{5}$, $\mathcal{M}_{2\!\times\!3}$     (e)  $\mathcal{M}_{2\!\times\!k}$, $\mathcal{M}_{k\!\times\!2}$

Which of these spaces are isomorphic to each other?

• (a)  $\mathbb{R}^{3}$     (b)  $\mathcal{M}_{2\!\times\!2}$     (c)  $\mathcal{P}_{3}$     (d)  $\mathbb{R}^{4}$     (e)  $\mathcal{P}_{2}$

Consider the isomorphism ${\rm Rep}_{B}(\cdot)\colon\mathcal{P}_{1}\to\mathbb{R}^{2}$ where $B=\langle 1,1+x\rangle$. Find the image of each of these elements of the domain.

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

For which $n$ is the space isomorphic to $\mathbb{R}^{n}$?

• $\mathcal{P}_{4}$

• $\mathcal{P}_{1}$

• $\mathcal{M}_{2\!\times\!3}$

• the plane $2x-y+z=0$ subset of $\mathbb{R}^{3}$

• the vector space of linear combinations of three letters $\{\hskip 1.00006ptax+by+cz\mid a,b,c\in\mathbb{R}\hskip 1.00006pt\}$

Show that if $m\neq n$ then $\mathbb{R}^{m}\not\cong\mathbb{R}^{n}$.

Is $\mathcal{M}_{m\!\times\!n}\cong\mathcal{M}_{n\!\times\!m}$?

Are any two planes through the origin in $\mathbb{R}^{3}$ isomorphic?

Find a set of equivalence class representatives other than the set of $\mathbb{R}^{n}$’s.

True or false: between any $n$-dimensional space and $\mathbb{R}^{n}$ there is exactly one isomorphism.

Can a vector space be isomorphic to one of its proper subspaces?

This subsection shows that for any isomorphism, the inverse map is also an isomorphism. This subsection also shows that for a fixed basis $B$ of an $n$-dimensional vector space $V$, the map $\text{Rep}_{B}\colon V\to\mathbb{R}^{n}$ is an isomorphism. Find the inverse of this map.

Prove these facts about matrices.

• The row space of a matrix is isomorphic to the column space of its transpose.

• The row space of a matrix is isomorphic to its column space.

Show that the function from Theorem 2.3 is well-defined.

Is the proof of Theorem 2.3 valid when $n=0$?

For each, decide if it is a set of isomorphism class representatives.

• $\{\hskip 1.00006pt\mathbb{C}^{k}\mid k\in\mathbb{N}\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt\mathcal{P}_{k}\mid k\in\text{$\{\hskip 1.00006pt-1,0,1,\ldots\hskip 1.00006pt\}$}\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt\mathcal{M}_{m\!\times\!n}\mid m,n\in\mathbb{N}\hskip 1.00006pt\}$

Let $f$ be a correspondence between vector spaces $V$ and $W$ (that is, a map that is one-to-one and onto). Show that the spaces $V$ and $W$ are isomorphic via $f$ if and only if there are bases $B\subset V$ and $D\subset W$ such that corresponding vectors have the same coordinates: ${\rm Rep}_{B}(\vec{v})={\rm Rep}_{D}(f(\vec{v}))$.

Consider the isomorphism $\text{Rep}_{B}\colon\mathcal{P}_{3}\to\mathbb{R}^{4}$.

• Vectors in a real space are orthogonal if and only if their dot product is zero. Give a definition of orthogonality for polynomials.

• The derivative of a member of $\mathcal{P}_{3}$ is in $\mathcal{P}_{3}$. Give a definition of the derivative of a vector in $\mathbb{R}^{4}$.

Does every correspondence between bases, when extended to the spaces, give an isomorphism? That is, suppose that $V$ is a vector space with basis $B=\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$ and that $f\colon B\to W$ is a correspondence such that $D=\langle f(\vec{\beta}_{1}),\ldots,f(\vec{\beta}_{n})\rangle$ is basis for $W$. Must $\hat{f}\colon V\to W$ sending $\vec{v}=c_{1}\vec{\beta}_{1}+\cdots+c_{n}\vec{\beta}_{n}$ to $\hat{f}(\vec{v})=c_{1}f(\vec{\beta}_{1})+\cdots+c_{n}f(\vec{\beta}_{n})$ be an isomorphism?

(Requires the subsection on Combining Subspaces, which is optional.) Suppose that $V=V_{1}\oplus V_{2}$ and that $V$ is isomorphic to the space $U$ under the map $f$. Show that $U=f(V_{1})\oplus f(V_{2})$.

Show that this is not a well-defined function from the rational numbers to the integers: with each fraction, associate the value of its numerator.