Show that w is in span(B). B= {-}~-B] We want to solve C₁ 18 2 0498 so set up the augmented matrix of the linear system and row-reduce to solve it: 3
Show that w is in span(B). B= {-}~-B] We want to solve C₁ 18 2 0498 so set up the augmented matrix of the linear system and row-reduce to solve it: 3
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
![**Linear Algebra - Coordinate Vectors**
### Problem Statement
**Show that** \( \mathbf{w} \) **is in the span of** \( \mathcal{B} \).
Given:
\[
\mathcal{B} = \left\{
\begin{pmatrix}
1 \\
2 \\
0
\end{pmatrix},
\begin{pmatrix}
1 \\
0 \\
-1
\end{pmatrix}
\right\}, \quad
\mathbf{w} =
\begin{pmatrix}
1 \\
8 \\
3
\end{pmatrix}
\]
### Solution Approach
We want to solve:
\[
c_1
\begin{pmatrix}
1 \\
2 \\
0
\end{pmatrix} +
c_2
\begin{pmatrix}
1 \\
0 \\
-1
\end{pmatrix} =
\begin{pmatrix}
1 \\
8 \\
3
\end{pmatrix}
\]
Set up the augmented matrix of the linear system and perform row reduction:
1. **Initial Matrix:**
\[
\begin{bmatrix}
1 & 1 & \vline & 1 \\
2 & 0 & \vline & 8 \\
0 & -1 & \vline & 3
\end{bmatrix}
\]
2. **Perform** \( R_2 - 2R_1 \):
\[
\begin{bmatrix}
1 & 1 & \vline & 1 \\
0 & -2 & \vline & 6 \\
0 & -1 & \vline & 3
\end{bmatrix}
\]
3. **Scale** \( -\frac{1}{2} R_2 \):
\[
\begin{bmatrix}
1 & 1 & \vline & 1 \\
0 & 1 & \vline & -3 \\
0 & -1 & \vline & 3
\end{bmatrix}
\]
4. **Row Operations** \( R_1 - R_2 \) and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6948d5d-782c-4f32-af22-abf47b4c3397%2Fcae2c997-f08f-4a7f-bcdc-e37e744a9f69%2F8732x6p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra - Coordinate Vectors**
### Problem Statement
**Show that** \( \mathbf{w} \) **is in the span of** \( \mathcal{B} \).
Given:
\[
\mathcal{B} = \left\{
\begin{pmatrix}
1 \\
2 \\
0
\end{pmatrix},
\begin{pmatrix}
1 \\
0 \\
-1
\end{pmatrix}
\right\}, \quad
\mathbf{w} =
\begin{pmatrix}
1 \\
8 \\
3
\end{pmatrix}
\]
### Solution Approach
We want to solve:
\[
c_1
\begin{pmatrix}
1 \\
2 \\
0
\end{pmatrix} +
c_2
\begin{pmatrix}
1 \\
0 \\
-1
\end{pmatrix} =
\begin{pmatrix}
1 \\
8 \\
3
\end{pmatrix}
\]
Set up the augmented matrix of the linear system and perform row reduction:
1. **Initial Matrix:**
\[
\begin{bmatrix}
1 & 1 & \vline & 1 \\
2 & 0 & \vline & 8 \\
0 & -1 & \vline & 3
\end{bmatrix}
\]
2. **Perform** \( R_2 - 2R_1 \):
\[
\begin{bmatrix}
1 & 1 & \vline & 1 \\
0 & -2 & \vline & 6 \\
0 & -1 & \vline & 3
\end{bmatrix}
\]
3. **Scale** \( -\frac{1}{2} R_2 \):
\[
\begin{bmatrix}
1 & 1 & \vline & 1 \\
0 & 1 & \vline & -3 \\
0 & -1 & \vline & 3
\end{bmatrix}
\]
4. **Row Operations** \( R_1 - R_2 \) and
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