Finding isomorphisms between vector spaces Find an isomorphism between the vector space of all $3 \times 3$ symmetric matrices and $\mathbb{R}^6$
Find two different isomorphisms between the vector space M22 of all $3 \times 3$ matrices and $\mathbb{R}^4$
Can someone please explain exactly what's going on here and what is being looked for?
 A: $\newcommand{\Reals}{\mathbf{R}}$In the hope that "a picture is worth a thousand words", here are two isomorphisms from the space of real, symmetric $2 \times 2$ matrices to $\Reals^{3}$, both viewed as real vector spaces with the standard operations:
$$
\left[\begin{array}{@{}cc@{}}
    a & b \\
    b & c \\
\end{array}\right] \mapsto \left[\begin{array}{@{}c@{}}
    a \\
    b \\
    c \\
  \end{array}\right],\qquad
\left[\begin{array}{@{}cc@{}}
    a & b \\
    b & c \\
\end{array}\right] \mapsto \left[\begin{array}{@{}c@{}}
    b \\
    a \\
    c \\
  \end{array}\right].
$$
(There are infinitely many others; in fact, the set of isomorphisms in this situation forms a $9$-parameter family.)

As for what's going on, an isomorphism of real vector spaces $V$ and $W$ is a bijective linear transformation $T:V \to W$. You should check (if it's not obvious) that the preceding maps are bijective linear transformations between the indicated spaces.
Conceptually, an isomorphism $T:V \to W$ assigns a "label" $w = T(v)$ to each element $v$ of $V$ in a "lossless" way: So far as questions of linear algebra are concerned, we may as well work in $W$ instead of $V$.
For example, if $T:V \to W$ is an isomorphism, $(v_{j})_{j=1}^{n}$ is an ordered set of vectors in $V$, and $w_{j} = T(v_{j})$, then:


*

*$(v_{j})$ spans $V$ if and only if $(w_{j})$ spans $W$;

*$(v_{j})$ is linearly independent if and only if $(w_{j})$ is lnearly independent.

*If $v = \sum_{j} c_{j} v_{j}$ for some scalars, then $w = T(v) = \sum_{j} c_{j} w_{j}$.
And so forth.
