# gram-schmidt (linear algebra)

Consider $A = [a_1, a_2, a_3] = $$\begin{bmatrix} 1 & 1 & 1 \\\ 1 & -1 & 0 \\\ 2 & 0 & 4 \\ \end{bmatrix} a) From the vectors a_1, a_2, a_3 find orthogonal vectors q_1, q_2, q_3 by Gram-Schmidt b) Find the upper triangular matrix R such that A = QR with Q = [q_1 q_2 q_3]. I believe I have found A_1 =$$ \begin{bmatrix} 1\\ 1\\ 2\\ \end{bmatrix}$, $A_2 = $$\begin{bmatrix} 1\\ -1\\ 0\\ \end{bmatrix} , and A_3 =$$ \begin{bmatrix} -1\\ -1\\ 1\\ \end{bmatrix}$, but I'm not sure where to go from there.

The vectors you found should be labeled $q_1,q_2,q_3$ instead of $A_1,A_2,A_3$. You have \begin{align} a_1 &= 1q_1+0q_2+0q_3 \\ a_2 &= 0q_1+1q_2+0q_3 \\ a_3 &= \frac{a_3^Tq_1}{q_1^Tq_1}q_1+\frac{a_3^Tq_2}{q_2^Tq_2}q_2+\frac{a_3^Tq_3}{q_3^Tq_3}q_3 \\ &= \frac{3}{2}q_1+\frac{1}{2}q_2+1q_3 \end{align} Therefore, $$\begin{bmatrix}a_1 & a_2 & a_3\end{bmatrix} = \begin{bmatrix}q_1 & q_1 & q_3\end{bmatrix}\begin{bmatrix}1 & 0 & \frac{3}{2} \\ 0 & 1 & \frac{1}{2} \\ 0 & 0 & 1\end{bmatrix}.$$