Let $A$ be an $m\times n$ matrix. Then under what conditions null space of $A$ and column space of $A$ may have common elements.
My attempt: I start by taking a vector $v\in \mathbb{R}^n$ such that $$v\in N(A)\cap Col(A)$$ where $N(A)$ is the column space of A and $Col(A)$ is column space of A. Then above means
$$v\in N(A)\implies Av=0, \ v\in Col(A)\implies v=Ay\ \text{for some}\ y.$$
Combining two we get $$A^2y=0.$$
Can I deduce anything from above? Please help.
Best Answer
Assume $A$ is square matrix and zero is an eigenvalue of $A$. If number of zero eigenvalues is $k$ and dimension of $N(A)$(also zero eigenspace) is strictly less than $k$, then we can directly say that $C(A) \cap N(A)$ has nontrivial element. It looks like the converse is true. But I could not prove yet.