Word Count : 451

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To find the QR decomposition of the given matrix \( A \), we'll follow these steps: \[ A = \begin{bmatrix}2 & -2 & 1\\2 & 2 & 1\\0 & 1 & 1\\1 & 0 & 1\end{bmatrix} \] ### Step 1: Apply the Gram-Schmidt Process We'll orthogonalize the columns of \( A \) to form an orthogonal matrix \( Q \). 1. First Column (\( \mathbf{a}_1 \)): \[ \mathbf{q}_1 = \frac{\mathbf{a}_1}{\|\mathbf{a}_1\|} = \frac{\begin{bmatrix}2\\2\\0\\1\end{bmatrix}}{\sqrt{2^2 + 2^2 + 0^2 + 1^2}} = \frac{\begin{bmatrix}2\\2\\0\\1\end{bmatrix}}{3} = \begin{bmatrix}\frac{2}{3}\\\frac{2}{3}\\0\\\frac{1}{3}\end{bmatrix} \] 2. Second Column (\( \mathbf{a}_2 \)): \[ \mathbf{v}_2 = \mathbf{a}_2 - (\mathbf{a}_2 \cdot \mathbf{q}_1)\mathbf{q}_1 \] \[ \mathbf{a}_2 \cdot \mathbf{q}_1 = (-2) \cdot \frac{2}{3} + 2 \cdot \frac{2}{3} + 1 _________ _________ _____ ________ _____ ______ _____ ___ ______ __________.
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