Find the vector x determined by the given coordinate vector [x]R and the given basis B. - 1 2 - 4 B= [x]g= 8 - 2 (Simplify your answer.)
Find the vector x determined by the given coordinate vector [x]R and the given basis B. - 1 2 - 4 B= [x]g= 8 - 2 (Simplify your answer.)
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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Question
![**Problem Statement:**
Find the vector **x** determined by the given coordinate vector \([x]_B\) and the given basis \(B\).
**Basis \(B\):**
\[
B = \left\{
\begin{bmatrix}
-1 \\
4 \\
0
\end{bmatrix},
\begin{bmatrix}
6 \\
-2 \\
6
\end{bmatrix},
\begin{bmatrix}
2 \\
-4 \\
5
\end{bmatrix}
\right\}
\]
**Coordinate Vector \([x]_B\):**
\[
[x]_B =
\begin{bmatrix}
-4 \\
8 \\
-2
\end{bmatrix}
\]
**Equation:**
\( \mathbf{x} = \text{(Simplify your answer.)} \)
**Solution Steps:**
To find the vector \(\mathbf{x}\), compute the linear combination of the basis vectors using the given coordinate vector components as coefficients:
\[
\mathbf{x} = -4 \begin{bmatrix} -1 \\ 4 \\ 0 \end{bmatrix} + 8 \begin{bmatrix} 6 \\ -2 \\ 6 \end{bmatrix} - 2 \begin{bmatrix} 2 \\ -4 \\ 5 \end{bmatrix}
\]
Simplify to find \(\mathbf{x}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75f96cc1-f20f-41e8-af87-e67129ee7f4c%2F7e6784fc-d5f8-4ae4-bcdf-88b6ecc6a714%2Faxw441_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the vector **x** determined by the given coordinate vector \([x]_B\) and the given basis \(B\).
**Basis \(B\):**
\[
B = \left\{
\begin{bmatrix}
-1 \\
4 \\
0
\end{bmatrix},
\begin{bmatrix}
6 \\
-2 \\
6
\end{bmatrix},
\begin{bmatrix}
2 \\
-4 \\
5
\end{bmatrix}
\right\}
\]
**Coordinate Vector \([x]_B\):**
\[
[x]_B =
\begin{bmatrix}
-4 \\
8 \\
-2
\end{bmatrix}
\]
**Equation:**
\( \mathbf{x} = \text{(Simplify your answer.)} \)
**Solution Steps:**
To find the vector \(\mathbf{x}\), compute the linear combination of the basis vectors using the given coordinate vector components as coefficients:
\[
\mathbf{x} = -4 \begin{bmatrix} -1 \\ 4 \\ 0 \end{bmatrix} + 8 \begin{bmatrix} 6 \\ -2 \\ 6 \end{bmatrix} - 2 \begin{bmatrix} 2 \\ -4 \\ 5 \end{bmatrix}
\]
Simplify to find \(\mathbf{x}\).
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