Exercise 5

Determine whether the following statements are true or false and explain your thinking.

1. If $A\xvec=\bvec$ is consistent, then $\xhat$ is a solution to $A\xvec=\bvec$.

2. If $R^2=1$, then the least-squares approximate solution $\xhat$ is also a solution to the original equation $A\xvec=\bvec$.

3. Given the $QR$ factorization $A=QR$, we have $A\xhat=Q^TQ\bvec$.

4. A $QR$ factorization provides a method for finding the least-squares approximate solution to $A\xvec=\bvec$ that is more reliable than solving the normal equations.

5. A solution to $AA^T\xvec = A\bvec$ is the least-squares approximate solution to $A\xvec = \bvec$.