column space and null space dimension of a matrix product I'm studying linear algebra on my own time. Came across the following problems:
The dimension of the column space of the matrix product $AB$


*

*same or greater than the dimension of the column space of $A$

*same or less than the dimension of the column space of $A$

*same or greater than the dimension of the column space of $B$

*same or less than the dimension of the column space of $B$


The dimension of the null space of the matrix product $AB$


*

*same or greater than the dimension of the column space of $A$

*same or less than the dimension of the column space of $A$

*same or greater than the dimension of the column space of $B$

*same or less than the dimension of the column space of $B$


How does one approach these problems? Would anyone care to help out?
 A: Here’s a way to think about this:
Recall that matrix multiplication corresponds to composition of linear transformations. That is, you can look at the product $ABx$ as feeding the vector $x$ first to the linear transformation represented by $B$, and then feeding the result of that to the linear transformation represented by $A$.  
The null space of a matrix is the set of vectors that its associated transformation maps to zero. The only possible image of the zero vector under a linear transformation is the zero vector of the codomain. So, if $B$ maps some vector $x$ to $0$, then $A$ can’t “unmap” that to something non-zero: once a vector gets sent to zero, it stays there. This means that the null space can’t get any smaller when you add another matrix to the chain of operations. (When I say “larger” and “smaller,” I mean dimension, not cardinality.)  
A similar line of reasoning can be applied to the column space of a product. The rank of a matrix—the dimension of its column space—gives you a maximum dimension for the image of a vector space under the associated linear transformation. In addition, you should be able to convince yourself that the dimension of that image can’t exceed the dimension of the space being mapped. So, the dimension of the column space can only stay the same or shrink at each stage.
