Exercise8

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

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 $$AB\text{.}$$

1. What are the dimensions of the product $$AB\text{?}$$

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

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

in-context