###### Exercise 8

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 \(AB\text{.}\)

What are the dimensions of the product \(AB\text{?}\)

Can you guarantee that the columns of \(AB\) span \(\real^3\text{?}\)

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 \(\real^3\text{?}\)