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The image shows a request to construct the QR decomposition of a given matrix. The matrix provided is:

\[
A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}
\]
This is a \( 4 \times 3 \) matrix.

### QR Decomposition Overview:
QR decomposition of a matrix \( A \) is a factorization of the matrix into an orthogonal matrix \( Q \) and an upper triangular matrix \( R \), such that:
\[
A = QR
\]
Where:
- \( Q \) is an orthogonal matrix (\( Q^T Q = I \)).
- \( R \) is an upper triangular matrix.

### Steps to Solve:
1. **Apply the Gram-Schmidt process** to the columns of \( A \) to obtain the orthogonal vectors that form the matrix \( Q \).
2. **Normalize** the orthogonal vectors to make them orthonormal, ensuring that the columns of \( Q \) are orthonormal.
3. The matrix \( R \) is then calculated by projecting the original columns of \( A \) onto the orthogonal vectors.

Let me now calculate the QR decomposition of the given matrix.

The QR decomposition of the matrix

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

results in:

1. The orthogonal matrix \( Q \) is:

\[
Q = \begin{bmatrix}
-0.5 & 0.866 & 0 \\
-0.5 & -0.289 & 0.816 \\
-0.5 & -0.289 & -0.408 \\
-0.5 & -0.289 & -0.408
\end{bmatrix}
\]

2. The upper triangular matrix \( R \) is:

\[
R = \begin{bmatrix}
-2 & -1.5 & -1 \\
0 & -0.866 & -0.577 \\
0 & 0 & -0.816
\end{bmatrix}
\]

This decomposition satisfies \( A = QR \).

Would you like further details on how to compute this step-by-step? Here are five related questions to consider:

1. What is the Gram-Schmidt process, and how is it used to form the \( Q \) matrix?
2. How do we determine if a matrix is orthogonal?
3. Can the QR decomposition be used to solve systems of linear equations?
4. How does the QR algorithm assist in finding eigenvalues of a matrix?
5. What are the applications of QR decomposition in numerical methods?

**Tip:** QR decomposition is especially useful for solving least squares problems where you need to minimize the error in approximations!

Construct the QR decomposition of the following matrix: 

 A = [
 [1, 0, 0], 
 [1, 1, 0], 
 [1, 1, 1], 
 [1, 1, 1] 
] (4x3 matrix)

Learn how to construct the QR decomposition of a 4x3 matrix using the Gram-Schmidt process. This guide covers key linear algebra concepts, including matrix factorization into orthogonal and upper triangular matrices. Suitable for university-level students.

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