Show that V is a subspace of M2x2 Matrices and Determine a basis A bit of information to start us off: Let V denote the set of all 2x2 matrices with equal column sums.

Show $V$ is a subspace of $M_{2\times 2}$ matrices:

and....

Determine a basis for $V$:

So for the first bit... $M$ is the set of all $2\times 2$ matrices, I used the definition of subspaces... A is a subset of B if for every element "a" in A, "a" is also in B. Am I able to then say, since $V$ is the set of all $2\times 2$ matrices with equal column sums, "$a$" being an element of $V$. Then "$a$" is a $2\times 2$ matrix. Since "$a$" is $2\times 2$ matrix, "$a$" is in $M_{2\times 2}$ as well. Therefore the set of all $2\times 2$ matrices with equal column sums is a subset of $M_{2\times2}$?

What I am really unsure of is how to determine a basis. I'm used to being given a spanning set and working from there. Thanks!
Note that a $2\times2$ matrix A has equal column sums iff $e^T A (e_1-e_2) = 0$, where $e=(1,1)^T$. The operator $L(A) = e^T A (e_1-e_2)$ is a linear function of $A$, hence $V = \ker L$, and so is a subspace (The kernel of a linear operator is automatically a subspace).

Since $\dim \mathbb{R}^{2 \times 2} = 4 = \dim \ker A + \dim {\cal R} A$, we see that $\dim V = \dim \ker A = 4-1 =3$.

Intuitively, note that if you pick 3 elements of a $2 \times 2$ matrix, you can always choose the fourth to satisfy the equation.

So, let the $A_{22}$ element be the 'special' element. A standard basis for $\mathbb{R}^{2 \times 2}$ is $e_i e_j^T$ for $i,j = 1,2$. So, pick one of these and figure out what the $22$ element must be to satisfy $L(A) = 0$.

Working through the computation gives $L(e_i e_j^T-\alpha e_2 e_2^T) = \delta_{1j}-\delta_{2j} + \alpha$, hence $\alpha = \begin{cases} -1 &amp; j=1 \\
+1 &amp; j = 2\end{cases}$. This construction produces three non-zero matrices that lie in $V$ by construction. It is easy to show that these are linearly independent, hence the form a basis.
More simplistically...

You must answer these three questions:


is $0$ (in this case, this means the zero matrix) in $V$?
if $A$ and $B$ are two matrices in V, is $A + B$ also in $V$?
if $r \in \mathbb{R}$ (or whatever field it is referring to, probably $\mathbb{R}$) and $A \in V$, then is $rA$ also in $V$?


The answer to the first question is yes (what are the column sums of the zero matrix?)

If the column sum of $A$ is $\alpha$ and the column sum of $B$ called $\beta$, what will the column sum of $A + B$ be?

Similarly, what will the column sum of $rA$ be?

For these questions, the "show it is a subspace" part is the easier part. Once you've got that, maybe try looking at some examples in your note for the basis part and try to piece it together from the other answer.