What is the $Q R$-decomposition of a matrix A? Explain how to construct the matrices $Q$ and $R$ by the Gram-Schmidt procedure, and show how the decomposition can be used to solve the matrix equation $A \mathbf{x}=\mathbf{b}$ when $A$ is a square matrix.

Why is this procedure not useful for numerical decomposition of large matrices? Give a brief description of an alternative procedure using Givens rotations.

Find a $Q R$-decomposition for the matrix

$$\mathrm{A}=\left[\begin{array}{rrrr} 3 & 4 & 7 & 13 \\ -6 & -8 & -8 & -12 \\ 3 & 4 & 7 & 11 \\ 0 & 2 & 5 & 7 \end{array}\right]$$

Is your decomposition unique? Use the decomposition you have found to solve the equation

$$A x=\left[\begin{array}{c} 4 \\ 6 \\ 2 \\ 9 \end{array}\right]$$