Find the coordinate vector (x)g of x relative to the given basis B= b, b2). (Simplify your answer.)
Find the coordinate vector (x)g of x relative to the given basis B= b, b2). (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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![Title: Finding the Coordinate Vector Relative to a Given Basis
Objective: Learn how to determine the coordinate vector of a given vector \( x \) relative to a specified basis \( B \).
Problem Statement:
Find the coordinate vector \([x]_B\) of \( x \) relative to the given basis \( B = \{ \mathbf{b_1}, \mathbf{b_2} \} \).
Given:
\[ \mathbf{b_1} = \begin{bmatrix} 1 \\ -2 \end{bmatrix}, \]
\[ \mathbf{b_2} = \begin{bmatrix} 2 \\ -3 \end{bmatrix}, \]
\[ x = \begin{bmatrix} 4 \\ -5 \end{bmatrix} \]
Solution:
Determine the coordinate vector \([x]_B\).
\[ [x]_B = \begin{bmatrix} \square \\ \square \end{bmatrix} \]
(Please simplify your answer.)
Explanation:
To find the coordinate vector \([x]_B\) of \( x \) relative to basis \( B \), solve the equation
\[ x = c_1 \mathbf{b_1} + c_2 \mathbf{b_2} \]
for \( c_1 \) and \( c_2 \). Here, \( c_1 \) and \( c_2 \) are the coordinates of \( x \) in the basis \( B \), which will be the components of the coordinate vector \([x]_B\).
In matrix form, this equation is:
\[ \begin{bmatrix} 4 \\ -5 \end{bmatrix} = c_1 \begin{bmatrix} 1 \\ -2 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ -3 \end{bmatrix} \]
By solving the above system for \( c_1 \) and \( c_2 \), we find the components of the coordinate vector \([x]_B\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8700fa5c-0cb9-46f9-817d-473369b2beec%2Fdb41c226-999e-455b-b69d-df1ae514bcc9%2F8yt54k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Finding the Coordinate Vector Relative to a Given Basis
Objective: Learn how to determine the coordinate vector of a given vector \( x \) relative to a specified basis \( B \).
Problem Statement:
Find the coordinate vector \([x]_B\) of \( x \) relative to the given basis \( B = \{ \mathbf{b_1}, \mathbf{b_2} \} \).
Given:
\[ \mathbf{b_1} = \begin{bmatrix} 1 \\ -2 \end{bmatrix}, \]
\[ \mathbf{b_2} = \begin{bmatrix} 2 \\ -3 \end{bmatrix}, \]
\[ x = \begin{bmatrix} 4 \\ -5 \end{bmatrix} \]
Solution:
Determine the coordinate vector \([x]_B\).
\[ [x]_B = \begin{bmatrix} \square \\ \square \end{bmatrix} \]
(Please simplify your answer.)
Explanation:
To find the coordinate vector \([x]_B\) of \( x \) relative to basis \( B \), solve the equation
\[ x = c_1 \mathbf{b_1} + c_2 \mathbf{b_2} \]
for \( c_1 \) and \( c_2 \). Here, \( c_1 \) and \( c_2 \) are the coordinates of \( x \) in the basis \( B \), which will be the components of the coordinate vector \([x]_B\).
In matrix form, this equation is:
\[ \begin{bmatrix} 4 \\ -5 \end{bmatrix} = c_1 \begin{bmatrix} 1 \\ -2 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ -3 \end{bmatrix} \]
By solving the above system for \( c_1 \) and \( c_2 \), we find the components of the coordinate vector \([x]_B\).
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