Prove that 2 vector spaces are isomorphic to each other Prove that the vector space V
$V = \begin{bmatrix}
 a&  0&  0&  0 \\
 b&  0&  0&  0 \\
 c&  d&  e&  f \\
\end{bmatrix}$
is isomorphic to the vector space $M_{2,3}$ of all 2 x 3 matrices under usual addition of matrices and scalar multiplication.
My prof didn't really explain as to how to go about this, but I was thinking that I could transform this into a one-dimensional vector, such that:
$V = (a, 0, 0, 0, b, 0, 0, 0, c, d, e, f)$
$M_{2,3}$ = (g, h, i, j, k, l)
I was thinking maybe I could remove the 0 values in V so they will have the same dimension? I'm not really sure. Please help.

The map
$\begin{pmatrix} 
a & 0 & 0 & 0 \\
b & 0 & 0 & 0 \\
c & d & e & f \\
\end{pmatrix} \rightarrow
\begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix}$
Is an isomorphism.

In general if you have clear bases for two vector spaces and they have the same dimension then you can give an explicit isomorfism in this way.
What I mean is that if $V$ has base $v_1,\dots, v_n$ and $U$ has base $u_1,\dots,u_n$ then the bijection $f(\sum\limits_{i=1}^n a_iv_i) = \sum\limits_{i=1}^na_iu_i$ is an isomorfism.
