Title: Finding the Coordinate Vector Relative to a Given BasisObjective: 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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Answered: Find the coordinate vector (x)g of x…

Math

Advanced Math

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

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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