Question

Orthonormalize the set ( eft{eft(begin{array}{l} 0end{array}right),lef\begin{array}{l}1 0 1end{array}right),left(\begin{array}{l}0 1 1end{array}rightright} )

Answer 1

The orthonormalized set is:
`\( \left\{ \begin{array}{l} \mathbf{u}_1 = \begin{bmatrix} 0 \\ (1)/(√(2)) \\ (1)/(√(2)) \end{bmatrix} \\ \mathbf{u}_2 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \\ \mathbf{u}_3 = \begin{bmatrix} 0 \\ (1)/(√(2)) \\ -(1)/(√(2)) \end{bmatrix} \end{array} \right\} \).`

Step-by-step explanation:

To orthonormalize the given set of vectors, we follow the steps of normalization and orthogonalization. First, we normalize each vector by dividing it by its magnitude. Let
`\( \mathbf{v}_1 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \), \( \mathbf{v}_2 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \), and \( \mathbf{v}_3 = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \).`

`\[ \mathbf{u}_1 = \frac{\mathbf{v}_1}{\|\mathbf{v}_1\|} = \begin{bmatrix} 0 \\ (1)/(√(2)) \\ (1)/(√(2)) \end{bmatrix} \]`

Next, we orthogonalize v2 and v3 with respect to u1:

`\[ \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2) = \frac{\langle \mathbf{v}_2, \mathbf{u}_1 \rangle}{\|\mathbf{u}_1\|^2} \cdot \mathbf{u}_1 = ((1)/(√(2)))/((1)/(2)) \cdot \begin{bmatrix} 0 \\ (1)/(√(2)) \\ (1)/(√(2)) \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \]`

`\[ \mathbf{u}_2 = \frac{\mathbf{v}_2 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2)}{\|\mathbf{v}_2 - \text{proj}_{\mathbf{u}_1}(\mathbf{v}_2)\|} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \]`

`\[ \text{proj}_{\mathbf{u}_1}(\mathbf{v}_3) = \frac{\langle \mathbf{v}_3, \mathbf{u}_1 \rangle}{\|\mathbf{u}_1\|^2} \cdot \mathbf{u}_1 = ((1)/(√(2)))/((1)/(2)) \cdot \begin{bmatrix} 0 \\ (1)/(√(2)) \\ (1)/(√(2)) \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \]`

`\[ \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3) = \frac{\langle \mathbf{v}_3, \mathbf{u}_2 \rangle}{\|\mathbf{u}_2\|^2} \cdot \mathbf{u}_2 = 0 \]`

Now, we normalize
`\( \mathbf{v}_3 - \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3) \):`

`\[ \mathbf{u}_3 = \frac{\mathbf{v}_3 - \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3)}{\|\mathbf{v}_3 - \text{proj}_{\mathbf{u}_2}(\mathbf{v}_3)\|} = \begin{bmatrix} 0 \\ (1)/(√(2)) \\ -(1)/(√(2)) \end{bmatrix} \]`

Thus, the orthonormalized set is
`\( \left\{ \begin{array}{l} \mathbf{u}_1 \\ \mathbf{u}_2 \\ \mathbf{u}_3 \end{array} \right\} \).`