From independent vectors $u_1 \cdots u_n$, Gram-Schmidt constructs orthonormal vectors $q_1 \cdots q_n$ .
The matrices with these columns satisfy $A=QR$ where R is upper triangular.
$$
A = QR \rightarrow Q = \begin{pmatrix} \vert & & \vert \\ \mathbf{q}_1 & \cdots & \mathbf{q}_n \\ \vert & & \vert \end{pmatrix}
$$
$$
R = Q^T A \quad \text{because} \quad Q^{-1} = Q^T
$$
#Exercises
#Exercise 1
- Find the $QR$ factorization of $A = \begin{pmatrix} 1 & 1 & 2 \ 1 & 1 & 0 \ 1 & 0 & 0 \end{pmatrix}$ $\mathbf{u}_1 = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix}, \mathbf{u}_2 = \begin{bmatrix} 1 \ 1 \ 0 \end{bmatrix}, \mathbf{u}_3 = \begin{bmatrix} 2 \ 0 \ 0 \end{bmatrix}$
#Exercise 2
- Find the $QR$ factorization of $A = \begin{pmatrix} 1 & 2 & 4 \ 0 & 0 & 5 \ 0 & 3 & 6 \end{pmatrix}$
Projection