Rank product of matrix compared to individual matrices. [duplicate]

Possible Duplicate:
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?

If $A$ is an $m\times n$ matrix and $B$ is a $n \times r$ matrix, prove that the rank of matrix $AB$ is at most $\mathrm{rank}(A)$.

I asked a similar question earlier phrased incorrectly. The above is closer to the actual question generalised.

Solution 1:

$rank(A)$ is the dimension of the column space of $A$. The product $Ab$, where $b$ is any column vector, is a column vector that lies in the column space of $A$. Therefore, all columns of $AB$ must be in the column space of $A$.

Solution 2:

The rank of $AB$ is equal to the dimension of the image of $AB,$ and similarly for the rank of $A.$ The image of $A$ contains the image of $AB.$