For the matrix above we're told that $A$ is a $4 \times 5$ matrix and that $\sigma_i > 0$ for all $i$. We're also told that $V^T \text{and } U$ are orthogonal matrices
We're asked to find a nonzero $X$ (if one exists) such that:
a)$AX = 0$ where X is a $5 \times 1$ matrix
b)$XA = 0$ where X is a $5 \times 4$ matrix
c)$AX = I_4$ where x is a $5 \times 4$ matrix
I'm confused about how to get $V^T \text{and } U$ since we don't know $A$. Here's my attempt for part (a). I feel like if I can get one of them, then that knowledge will transfer really well to the other parts.
We know that the last column of $V^T$ is in the null space of A by definition of the SVD. We are guaranteed to have a null space of rank $1$, since we have a $\sigma = 0$. How do I go about finding this vector in terms of $U, V, \text{and } \sigma_i$.
Is there some fact I'm missing about SVD's?