1.
Solution.
Since is invertible, we know that we can find its inverse by row reducing the augmented matrix
In particular, this says that the RREF form of is
One way to finish is that the information above says that has only the trivial solution, which means by Observation 1.3.2 that the columns of are linearly independent. Since there are of them, by Theorem 3.1.6, they must be a basis.
Another approach is that the linear map given by is an isomorphism with the inverse map being given In particular, is surjective and its image is the column space of That means that the columns of span all of and hence must be a basis again by Theorem 3.1.6.