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given two matrices $A,B$ of order $m\times n$ and $n\times p$ respectively, could anyone help me which one is true and false by counter example?

(1) every column of $AB$ is a linear combination of column of $A$

(2) every column of $AB$ is a linear combination of column of $B$

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    I suggest you try a couple of examples using 2 x 2 matrices. It will help if A is a singular matrix in one example and B is singular in the other. – 2017-01-25

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Think of the matrices as linear transformation under suitable basis.

The columns of $A$ span the range of $A$, the columns of $AB$ span the range of $A\circ B$.

Since the range of $A\circ B$ is clearly contained in the range of $A$ we have that every column of $AB$ is a linear combination of columns of $A$.

Clearly the second statement is false as the columns don't even need to have the same size.