Matrix vector spaces isomorphic to column vector spaces? my question is a basic linear algebra question, so hopefully someone can answer without too much trouble.
My question was motivated by a problem I was doing about a linear transformation from the vector space V of all real 2x3 matrices onto the vector space W of 4x1 column vectors.
I guessed that the dimension of V was 6, and was wondering then if V was isomorphic to the vector space of 6x1 column vectors, call it C. I'm sure it is NOT isomorphic, but they have the same dimension, so perhaps V could be isomorphic to a vector space of 6x1 column vectors that satisfy certain conditions? or maybe V would have to satisfy certain conditions?
Could someone help me clarify my thoughts on this? Thanks!
 A: Every finite dimensional vector space can be identified with a space of column vectors with the same dimension. For concreteness we can consider a 4 dimensional vector space. Since this space is four dimensional we know that we can find four linearly independent vectors and use them as a basis on the space. 
$\mathbb{V}$ is a 4 dimensional vector space. 
Let $\vec{e_1},\vec{e_2},\vec{e_3},\vec{e_4} \in \mathbb{V}$ be a basis on $\mathbb{V}$. 
Then every vector $v \in \mathbb{V}$ can be written as: $\vec{v} = v_1 \vec{e_1} + v_2 \vec{e_2} + v_3 \vec{e_3} + v_4 \vec{e_4}$ where the coefficients $v_1,v_2,v_3,v_4$ are scalars (typically complex numbers). 
This means that every vector in the space can be uniquely identified with its coefficients when expressed in the chosen basis.
For each $\vec{v} \in \mathbb{V}$ we can identify $\vec{v} \equiv (v_1,v_2,v_3,v_4)$.
You can verify that when we add vectors the corresponding coefficients add and obey all the other rules required of a vector space. The coefficients can be regarded as the elements of a column vector.
In practice you typically try to do this when working with  generic vector space. Identifying a vector space with a space of column vectors helps you to leverage the intuition you already have developed for column vectors. However there are some notable exceptions in the context of infinite dimensional vector spaces where you cannot define such an isomorphism.
