I have a subspace $W$, which has a dimension of $3$. I am also familiar with the base of this subspace. Now, I need to find the subspace $U$ so that $W + U= M_{3x3}$ is a direct sum. The approach that I was thinking of is to use the base of $W$ and extend it to the base of $M_{3x3}$. After I did that, the vectors from this extension would form the base of the unknown subspace $U$.
Hope I made the question clear enough. Just wondering if this approach is correct and would lead me to the correct answer. Thanks :)