Exercise 6

Suppose that $A$ is a matrix with linearly independent columns and having the factorization $A=QR$. Determine whether the following statements are true or false and explain your thinking.

1. It follows that $R=Q^TA$.

2. The matrix $R$ is invertible.

3. The product $Q^TQ$ projects vectors orthogonally onto $\col(A)$.

4. The columns of $Q$ are an orthogonal basis for $\col(A)$.

5. The orthogonal complement $\col(A)^\perp = \nul(Q^T)$.