Question

Let A and B be matrices in the plane spanned by the columns of C. Is AB also in the plane spanned by the columns of C? Why or why not?

Answer 1

If A and B are matrices in the plane spanned by the columns of C, then AB is also in the plane spanned by the columns of C.

Step-by-step explanation:

In this case, let's assume that the columns of matrix C are represented by vectors v1 and v2. So, C = [v1, v2].

We are given that A and B are matrices in the plane spanned by the columns of C. This means that A and B can be represented as linear combinations of v1 and v2.

If A = x1v1 + x2v2 and B = y1v1 + y2v2, then AB = (x1v1 + x2v2)(y1v1 + y2v2) = x1y1v1 + x1y2v2 + x2y1v1 + x2y2v2.

This implies that AB can also be represented as a linear combination of v1 and v2, which means that AB is also in the plane spanned by the columns of C.