Exercise 8

The following observation will be helpful in this exercise. If we want to find a solution to the equation $AB\xvec = \bvec$, we could first find a solution to the equation $A\yvec = \bvec$ and then find a solution to the equation $B\xvec = \yvec$.

Suppose that $A$ is a $3\times 4$ matrix whose columns span $\real^3$ and $B$ is a $4\times 5$ matrix. In this case, we can form the product $AB$.

1. What is the shape of the product $AB$?

2. Can you guarantee that the columns of $AB$ span $\real^3$?

3. If you know additionally that the span of the columns of $B$ is $\real^4$, can you guarantee that the columns of $AB$ span $\real^3$?