Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Statement:**
The vector **x** is in a subspace **H** with a basis **B** = {**b₁**, **b₂**}. Find the B-coordinate vector of **x**.
**Given:**
\[
\mathbf{b}_1 =
\begin{bmatrix}
5 \\
-7
\end{bmatrix}, \quad
\mathbf{b}_2 =
\begin{bmatrix}
-1 \\
3
\end{bmatrix}, \quad
\mathbf{x} =
\begin{bmatrix}
5 \\
1
\end{bmatrix}
\]
**Required:**
\[
[\mathbf{x}]_B = \, \_
\]
---
**Explanation:**
To find the B-coordinate vector [**x**]_B, we need to express vector **x** as a linear combination of the basis vectors **b₁** and **b₂**. That is, we need to find scalars \(c_1\) and \(c_2\) such that:
\[
\mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2
\]
Substituting the given vectors, we have:
\[
\begin{bmatrix}
5 \\
1
\end{bmatrix} = c_1
\begin{bmatrix}
5 \\
-7
\end{bmatrix} + c_2
\begin{bmatrix}
-1 \\
3
\end{bmatrix}
\]
This equation represents a system of linear equations:
\[
5c_1 - c_2 = 5
\]
\[
-7c_1 + 3c_2 = 1
\]
Solving this system using methods such as substitution, elimination, or matrix operations will give the values of \(c_1\) and \(c_2\). These values constitute the components of the B-coordinate vector [**x**]_B.
---
**Graphical/Diagram Explanation:**
Given the problem statement and equations, this exercise falls under the topic of vector spaces and coordinate systems in Linear Algebra. The solution requires solving a system of linear equations which would typically involve using matrix techniques such as Gaussian elimination.](https://content.bartleby.com/qna-images/question/1bfda259-7429-4717-b69c-7067c249bba0/c7ab489e-5132-4b1d-bcc8-b4168a8de02e/xd0a3cm_processed.png)
Transcribed Image Text:**Problem Statement:**
The vector **x** is in a subspace **H** with a basis **B** = {**b₁**, **b₂**}. Find the B-coordinate vector of **x**.
**Given:**
\[
\mathbf{b}_1 =
\begin{bmatrix}
5 \\
-7
\end{bmatrix}, \quad
\mathbf{b}_2 =
\begin{bmatrix}
-1 \\
3
\end{bmatrix}, \quad
\mathbf{x} =
\begin{bmatrix}
5 \\
1
\end{bmatrix}
\]
**Required:**
\[
[\mathbf{x}]_B = \, \_
\]
---
**Explanation:**
To find the B-coordinate vector [**x**]_B, we need to express vector **x** as a linear combination of the basis vectors **b₁** and **b₂**. That is, we need to find scalars \(c_1\) and \(c_2\) such that:
\[
\mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2
\]
Substituting the given vectors, we have:
\[
\begin{bmatrix}
5 \\
1
\end{bmatrix} = c_1
\begin{bmatrix}
5 \\
-7
\end{bmatrix} + c_2
\begin{bmatrix}
-1 \\
3
\end{bmatrix}
\]
This equation represents a system of linear equations:
\[
5c_1 - c_2 = 5
\]
\[
-7c_1 + 3c_2 = 1
\]
Solving this system using methods such as substitution, elimination, or matrix operations will give the values of \(c_1\) and \(c_2\). These values constitute the components of the B-coordinate vector [**x**]_B.
---
**Graphical/Diagram Explanation:**
Given the problem statement and equations, this exercise falls under the topic of vector spaces and coordinate systems in Linear Algebra. The solution requires solving a system of linear equations which would typically involve using matrix techniques such as Gaussian elimination.
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