Let $A$ be an $m \times n$ matrix and $B$ be an $n \times p$ matrix.
- Prove that $null{AB} \subseteq null{B}$.
- Give an example of $A$ and $B$ such that $null{AB} \not\supseteq null{A}$.
- Prove or disprove: $CS(AB) \supseteq CS(B)$.
- Suppose $m = n = p$, that $\det(B) \neq 0$ and there are $k$ linearly independent vectors in $null{A}$ for some $k \leq n$. Show that there are $k$ linearly independent vectors in $null{AB}$.
My approaches: I set up $x$ in $null(B)$, then $Bx = 0$, $ABx = A(Bx) = A(0) = 0 $ So $x$ in $null(AB)$. So, $null(AB) \subseteq null(B) $.
I need help on the proofs... thanks