We use the Gram-Schmidt process to convert the column vectors of A into a set of orthogonal vectors u_1, u_2, u_3. The projection formula proj_u v = (v · u)/(u · u) u is applied iteratively. For u_2, we subtract the projection of a_2 onto u_1. For u_3, we subtract the projections of a_3 onto both u_1 and u_2.

We use the Gram-Schmidt process to convert the column vectors of A into a set of orthogonal vectors u_1, u_2, u_3. The projection formula proj_u v = (v · u)/(u · u) u is applied iteratively. For u_2, we subtract the projection of a_2 onto u_1. For u_3, we subtract the projections of a_3 onto both u_1 and u_2.

To form the matrix Q, we normalize each orthogonal vector u_i by dividing it by its magnitude |u_i|. This results in a set of orthonormal vectors q_i, which become the columns of the orthogonal matrix Q.

To form the matrix Q, we normalize each orthogonal vector u_i by dividing it by its magnitude |u_i|. This results in a set of orthonormal vectors q_i, which become the columns of the orthogonal matrix Q.

Find the QR decomposition of $A = [[1, 1, 0], [1, 0, 1], [0, 1, 1]]$ .

Answer: $Q = \begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{6} & -1/\sqrt{3} \\ 1/\sqrt{2} & -1/\sqrt{6} & 1/\sqrt{3} \\ 0 & 2/\sqrt{6} & 1/\sqrt{3} \end{bmatrix}, R = \begin{bmatrix} \sqrt{2} & 1/\sqrt{2} & 1/\sqrt{2} \\ 0 & 3/\sqrt{6} & 1/\sqrt{6} \\ 0 & 0 & 2/\sqrt{3} \end{bmatrix}$

Step-by-step solution

Step 1: Define the columns of A

We begin by representing the given matrix $A$ in terms of its column vectors. This is the first step in applying the Gram-Schmidt orthogonalization process to find the QR decomposition.

Step 2: Apply Gram-Schmidt to find orthogonal vectors

We use the Gram-Schmidt process to convert the column vectors of $A$ into a set of orthogonal vectors $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ . The projection formula $\text{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$ is applied iteratively. For $\mathbf{u}_2$ , we subtract the projection of $\mathbf{a}_2$ onto $\mathbf{u}_1$ . For $\mathbf{u}_3$ , we subtract the projections of $\mathbf{a}_3$ onto both $\mathbf{u}_1$ and $\mathbf{u}_2$ .

Step 3: Normalize orthogonal vectors to find Q

To form the matrix $Q$ , we normalize each orthogonal vector $\mathbf{u}_i$ by dividing it by its magnitude $\|\mathbf{u}_i\|$ . This results in a set of orthonormal vectors $\mathbf{q}_i$ , which become the columns of the orthogonal matrix $Q$ .

Step 4: Calculate R using $R = Q^T A$

The matrix $R$ can be found by multiplying the transpose of $Q$ with the original matrix $A$ , i.e., $R = Q^T A$ . Since $Q$ is an orthogonal matrix, $Q^T Q = I$ , and thus $A = QR$ implies $Q^T A = Q^T QR = IR = R$ . This calculation yields an upper triangular matrix $R$ .

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Find the QR decomposition of A=[[1,1,0],[1,0,1],[0,1,1]]A = [[1, 1, 0], [1, 0, 1], [0, 1, 1]]A=[[1,1,0],[1,0,1],[0,1,1]].