= √1 = Let 4] 2 V2 - 2 and 3 = 2 -7 Explain that B = (V1, V2, V3) forms a basis of a 3-dimensional space. Find [u] (the coordinate of u with respect to the basis B) by writing the vector u as a linear combination of 7₁, 72, and 3.
= √1 = Let 4] 2 V2 - 2 and 3 = 2 -7 Explain that B = (V1, V2, V3) forms a basis of a 3-dimensional space. Find [u] (the coordinate of u with respect to the basis B) by writing the vector u as a linear combination of 7₁, 72, and 3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
![Let
\[
\mathbf{u} = \begin{bmatrix} 7 \\ 6 \\ 7 \end{bmatrix}, \quad \mathbf{v}_1 = \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 3 \\ 2 \\ -7 \end{bmatrix}, \quad \text{and} \quad \mathbf{v}_3 = \begin{bmatrix} -2 \\ 1 \\ 2 \end{bmatrix}.
\]
Explain that \(\mathcal{B} = (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)\) forms a basis of a 3-dimensional space. Find \([\mathbf{u}]_{\mathcal{B}}\) (the coordinate of \(\mathbf{u}\) with respect to the basis \(\mathcal{B}\)) by writing the vector \(\mathbf{u}\) as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \text{and} \mathbf{v}_3\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ef06bb3-2d9b-4f27-bb3b-835b443ab608%2F9e9b7c0f-53e9-485e-b8e5-44ddb5395896%2F7f7ejh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let
\[
\mathbf{u} = \begin{bmatrix} 7 \\ 6 \\ 7 \end{bmatrix}, \quad \mathbf{v}_1 = \begin{bmatrix} 3 \\ 4 \\ -1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 3 \\ 2 \\ -7 \end{bmatrix}, \quad \text{and} \quad \mathbf{v}_3 = \begin{bmatrix} -2 \\ 1 \\ 2 \end{bmatrix}.
\]
Explain that \(\mathcal{B} = (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)\) forms a basis of a 3-dimensional space. Find \([\mathbf{u}]_{\mathcal{B}}\) (the coordinate of \(\mathbf{u}\) with respect to the basis \(\mathcal{B}\)) by writing the vector \(\mathbf{u}\) as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \text{and} \mathbf{v}_3\).
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