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: