Let $W$ be the null space of the matrix.
$\begin{pmatrix}1& 0 & 2 & 0 & 1\\ 0& 0 &0 & 1 & 3\\ 0 &1 &3 &0 &2\\ 0 & 0 & 0 & 0 & 1\end{pmatrix}$.
What is the dimension of $W^\perp$?
The answer I was given is 4, but I don't know why.
I know the rank of the matrix is 3, so the nullity is 2. The nullity is the dimension of the nullspace, so that is $W$ in this problem.
I also know that $dim(V) = dim(W) + dim(W^\perp)$, but I don't know the dimension of V.
Am I even going in the right direction? Any help is appreciated.
If you switch the second and third row, the matrix become row echelon form, and we can see that the rank of the matrix is $4$ since we have $4$ pivot columns. You should be able to adjust the nullity accordingly.
Elements of nullspace lives in $\mathbb{R}^5$, $V=\mathbb{R}^5$.