Exercise 8

Suppose we have the $QR$ factorization $A=QR$ where the columns of $A$ are $\avec_1,\avec_2,\ldots,\avec_n$ and the columns of $R$ are $\rvec_1,\rvec_2,\ldots,\rvec_n$.

1. How can the matrix product $A^TA$ be expressed in terms of dot products?

2. How can the matrix product $R^TR$ be expressed in terms of dot products?

3. Explain why $A^TA=R^TR$.

4. Explain why the dot products $\avec_i\cdot\avec_j = \rvec_i\cdot\rvec_j$.