Q:

Linear dependence of the columns of $AB$?

If $A_{m \times n}$ and $B_{n \times p}$ are matrices in which the columns of $B$ are linearly dependent, then how to prove that the columns of $AB$ are linearly dependent.
I can see them multiplying but I was trying to bring them in the form of a linear combination after which I can apply the linear dependence in which atleast one of the coefficients are not zero.

A:

Let $b_i$ be the columns of $B$.
If we can find coefficients $c_i$'s where at least one of them is non-zero $$\sum_{i=1}^n c_ib_i=0$$ 
We just have to premultiply $A$ and you should obtain a relationship among columns of $AB$. 

