find the QR factorization of A=[1 -1 1 ; 2 0 -1 ; 0 1 1] 

To find the QR factorization of the matrix \( A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 0 & -1 \\ 0 & 1 & 1 \end{bmatrix} \), we need to decompose \( A \) into the product of an orthogonal matrix \( Q \) and an upper triangular matrix \( R \).

### Step 1: Gram-Schmidt Process

We use the Gram-Schmidt process to generate the orthonormal vectors (columns of \( Q \)) from the columns of \( A \).

Let the columns of \( A \) be \( \mathbf{a}_1 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix} \), \( \mathbf{a}_2 = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} \), and \( \mathbf{a}_3 = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} \).

#### Step 1.1: First vector \( \mathbf{q}_1 \)

Normalize \( \mathbf{a}_1 \):
\[
\mathbf{q}_1 = \frac{\mathbf{a}_1}{\|\mathbf{a}_1\|}
= \frac{1}{\sqrt{1^2 + 2^2 + 0^2}} \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}
= \frac{1}{\sqrt{5}} \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}
\]

#### Step 1.2: Second vector \( \mathbf{q}_2 \)

First, project \( \mathbf{a}_2 \) onto \( \mathbf{q}_1 \):
\[
\text{proj}_{\mathbf{q}_1} \mathbf{a}_2 = \frac{\mathbf{q}_1^T \mathbf{a}_2}{\mathbf{q}_1^T \mathbf{q}_1} \mathbf{q}_1
= \frac{\frac{1}{\sqrt{5}} \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}^T \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}}{1} \mathbf{q}_1
= \frac{1}{\sqrt{5}} \cdot (-1 + 0) \mathbf{q}_1
= -\frac{1}{\sqrt{5}} \mathbf{q}_1
\]

Now subtract the projection from \( \mathbf{a}_2 \) to find the orthogonal component:
\[
\mathbf{u}_2 = \mathbf{a}_2 - \text{proj}_{\mathbf{q}_1} \mathbf{a}_2
= \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} - \left( -\frac{1}{\sqrt{5}} \right) \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}
= \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} + \frac{1}{\sqrt{5}} \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}
= \begin{bmatrix} -1 + \frac{1}{\sqrt{5}} \\ 0 + \frac{2}{\sqrt{5}} \\ 1 \end{bmatrix}
= \begin{bmatrix} -\frac{5}{\sqrt{5}} + \frac{1}{\ ans

find the QR factorization of A=[1 -1 1 ; 2 0 -1 ; 0 1 1]

This page provides a detailed step-by-step guide on how to find the QR factorization of matrix A = [1 -1 1; 2 0 -1; 0 1 1] using the Gram-Schmidt process. It covers key concepts of linear algebra, including orthogonalization, normalization, and projection, suitable for undergraduate students.

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