If we're given that $u ∈ R ^{m×1} , a ∈ R ^{m×1} , v ∈ R ^{n×1} ,$ and $b ∈ R ^{n×1}$
and a matrix $B$ given by $B = uv^T + ab^T$.
What relationship between u, v, a, and b would make matrix $B$ equal to $0$? I'm confused as to how to approach this problem.
You can look at it component-wise. We get:
$$B_{ij} = u_iv_j + a_ib_j$$
Where $1 \leq i \leq m$ and $1 \leq j \leq n$. So the equation is $u_iv_j + a_ib_j = 0$.