Find the QR decomposition of the matrix

To find the QR decomposition of a matrix, we decompose it into an orthogonal matrix \( Q \) and an upper triangular matrix \( R \). The QR decomposition of a matrix \( A \) can be represented as \( A = QR \).

We can use the Gram-Schmidt process to find the QR decomposition.

Given the matrix:

\[ A = \begin{bmatrix} 2 & -2 & 1\\ 2 & 2 & 1\\ 0& 1& 1\\ 1 & 0 & 1 \end{bmatrix} \]

We start by normalizing the columns of \( A \) to obtain an orthogonal matrix \( Q \):

1. Normalize the first column:
\[ v_1 = \begin{bmatrix} 2 \\ 2 \\ 0 \\ 1 \end{bmatrix} \]
\[ q_1 = \frac{v_1}{\|v_1\|} = \frac{1}{\sqrt{10}} \begin{bmatrix} 2 \\ 2 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{2}{\sqrt{10}} \\ \frac{2}{\sqrt{10}} \\ 0 \\ \frac{1}{\sqrt{10}} \end{bmatrix} \]

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