Let B={(1,-2,1), (4,-7,5), (5,-8,8)], and x=(-6,10,-7) Find [x]B
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:**
Let \( B = \{ (1, -2, 1), (4, -7, 5), (5, -8, 8) \} \) and \( x = (-6, 10, -7) \).
Find \( [x]_B \).
*Instructions:*
Give your answer in the form \( (a,b,c) \) with no spaces.
---
**Explanation:**
In this problem, you are given a set \( B \) of vectors and a vector \( x \). The goal is to find the coordinates of \( x \) with respect to the basis \( B \). This process involves solving a system of linear equations to express \( x \) as a linear combination of the vectors in \( B \).
To find the coordinates \( [x]_B \), follow these steps:
1. **Set up the equation**: Express \( x \) as a linear combination of the basis vectors, i.e., find \( a, b, \) and \( c \) such that:
\[
x = a(1, -2, 1) + b(4, -7, 5) + c(5, -8, 8)
\]
2. **Form the system of equations** from the above vector equation by comparing the respective components:
\[
\begin{cases}
a + 4b + 5c = -6 \\
-2a - 7b - 8c = 10 \\
a + 5b + 8c = -7
\end{cases}
\]
3. **Solve the system of equations** to find \( a, b, \) and \( c \).
4. **Provide the solution** in the form \( (a,b,c) \) with no spaces.
Good luck!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5483afab-7850-40f6-959f-e698dea1419a%2F16b771e7-1b4f-47b4-b09c-fb84a127e101%2Ffo6y5gm_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( B = \{ (1, -2, 1), (4, -7, 5), (5, -8, 8) \} \) and \( x = (-6, 10, -7) \).
Find \( [x]_B \).
*Instructions:*
Give your answer in the form \( (a,b,c) \) with no spaces.
---
**Explanation:**
In this problem, you are given a set \( B \) of vectors and a vector \( x \). The goal is to find the coordinates of \( x \) with respect to the basis \( B \). This process involves solving a system of linear equations to express \( x \) as a linear combination of the vectors in \( B \).
To find the coordinates \( [x]_B \), follow these steps:
1. **Set up the equation**: Express \( x \) as a linear combination of the basis vectors, i.e., find \( a, b, \) and \( c \) such that:
\[
x = a(1, -2, 1) + b(4, -7, 5) + c(5, -8, 8)
\]
2. **Form the system of equations** from the above vector equation by comparing the respective components:
\[
\begin{cases}
a + 4b + 5c = -6 \\
-2a - 7b - 8c = 10 \\
a + 5b + 8c = -7
\end{cases}
\]
3. **Solve the system of equations** to find \( a, b, \) and \( c \).
4. **Provide the solution** in the form \( (a,b,c) \) with no spaces.
Good luck!
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