Specifically, I'm trying to solve the following problem:
Let $V=M_3(\mathbb{R})$ be the real vector space of all $3\times 3$ real matrices. Consider the subspace $$W=\left\{\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\in V:a_{i1}+a_{i2}+a_{i3}=a_{1i}+a_{2i}+a_{3i}=0\text{ for }i=1,2,3\right\}.$$ Find an explicit basis for $W$ and an explicit basis for a subspace $U\leq V$ satisfying $V=W\oplus U$.
The first part is not too hard: Observe that any element of $W$ has the form $\begin{pmatrix}a&b&-a-b\\c&d&-c-d\\-a-c&-b-d&a+b+c+d\end{pmatrix}$, where $a,b,c,d\in\mathbb{R}$. It is easily verified that $$\left\{\begin{pmatrix}1&0&-1\\0&0&0\\-1&0&1\end{pmatrix},\begin{pmatrix}0&1&-1\\0&0&0\\0&-1&1\end{pmatrix},\begin{pmatrix}0&0&0\\1&0&-1\\-1&0&1\end{pmatrix},\begin{pmatrix}0&0&0\\0&1&-1\\0&-1&1\end{pmatrix}\right\}$$ is a basis for $W$.
The second part is where I'm stuck. I need to construct a basis for a space $U$ that trivially intersects $W$, but sums with $W$ to all of $V$. So $W$ is the space of all $3\times3$ matrices with row/column sum equal to $0$, and $U$ is the space of all $3\times3$ matrices with row/column sum not equal to $0$.
I see this is much easier than I was thinking. One way to do it is with $\{E_{31},E_{32},E_{33},E_{23},E_{13}\}$. I.e. just take the basis for $W$ and extend it to a basis for $V$.