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1 Consider the vectors vị = For these vectors: (a) Show that the set of these three vectors is an orthogonal basis for IR³. (b) Express x = 2 as a linear combination of v1, V2, V3 using Theorem 5.2. (c) Determine if {V1, v2, V3} is an orthonormal set. If not, normalize the vectors to form an orthonormal set.

1 Consider the vectors vị = For these vectors: (a) Show that the set of these three vectors is an orthogonal basis for IR³. (b) Express x = 2 as a linear combination of v1, V2, V3 using Theorem 5.2. (c) Determine if {V1, v2, V3} is an orthonormal set. If not, normalize the vectors to form an orthonormal set.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the vectors \(\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\), \(\mathbf{v}_2 = \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}\), \(\mathbf{v}_3 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}\). For these vectors: (a) Show that the set of these three vectors is an orthogonal basis for \(\mathbb{R}^3\). (b) Express \(\mathbf{x} = \begin{bmatrix} 4 \\ 2 \\ 3 \end{bmatrix}\) as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) using Theorem 5.2. (c) Determine if \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is an orthonormal set. If not, normalize the vectors to form an orthonormal set.
Transcribed Image Text:Consider the vectors \(\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\), \(\mathbf{v}_2 = \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix}\), \(\mathbf{v}_3 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}\). For these vectors: (a) Show that the set of these three vectors is an orthogonal basis for \(\mathbb{R}^3\). (b) Express \(\mathbf{x} = \begin{bmatrix} 4 \\ 2 \\ 3 \end{bmatrix}\) as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) using Theorem 5.2. (c) Determine if \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is an orthonormal set. If not, normalize the vectors to form an orthonormal set.
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