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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.

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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, $$ \begin{bmatrix}a_1 & a_2 & a_3\end{bmatrix} = \begin{bmatrix}q_1 & q_1 & q_3\end{bmatrix}\begin{bmatrix}1 & 0 & \frac{3}{2} \\ 0 & 1 & \frac{1}{2} \\ 0 & 0 & 1\end{bmatrix}. $$

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