Question

Suppose A=QR, where Q is mxn and Ris nxn Show that if the columns of A are linearly independent, then R must be invertible. [Hint Study the equation Rx = 0 and use the fact that A=QR] The matrix R is _________

Answer 1

If the columns of matrix A are linearly independent, then matrix R is invertible.

Step-by-step explanation:

If we suppose A=QR, where Q is an mxn matrix and R is an nxn matrix, and the columns of A are linearly independent, we can show that R must be invertible.

To understand why, consider the equation Rx = 0.

If R were not invertible, it would mean that it has a non-trivial null space, which implies that there exists a non-zero vector x such that Rx=0.

However, since A=QR and A's columns are linearly independent, it means that the only solution to the equation Q(Rx) = Q(0) is x=0, indicating that R has a trivial null space and therefore is invertible.

Thus, the matrix R is invertible.