Column and Nullspace of matrix AB

Question

Where $A$ is a $5 x 4$ matrix and $B$ is a $4 x 3$ matrix.

1. If $w$ is in $Nul(B)$ then $w$ is in $Nul(AB)$
2. If $x$ is in $Nul(A)$ then $x$ is not in $Nul(AB)$
3. If $v$ is in $Col(A)$, then $v$ is not in $Col(AB)$
4. If $y$ is in $Col(B)$, then $y$ is in $Col(AB)$

For me, it is unclear how I should be able to prove these statements how these are valid or invalid. I believe all the statements to be valid, but I am not sure. How should I approach this problem?

Answer 1

Most of them are direct consequence of the definitions. For example,

1. For 1: if $w \in \text{Null}(B)$, then $Bw=0$, thus $A(Bw)=A(0)=0$, so $AB(w)=0$. This means $w \in \text{Null}(AB)$.
2. For 2: Since $A$ has size $5 \times 4$, therefore the $\text{Null}(A) \subset \Bbb{R}^4$ (assuming all the matrices are over real numbers). This means if $w \in \text{Null}(A)$ then $w$ must be a $4-$dimensional vector (i.e. $w$ is represented as $4 \times 1$ column vector). In which case $Bw$ is not even defined, hence $w \not\in \text{Null}(AB)$.

I hope you can now proceed with the other two as well.

Hint: be cautious with #3, think about the zero vector. (I am assuming you have a typo and both the vectors are $v$)