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Find the coordinate vector [x]g of x relative to the given basis B = {b₁,b₂, b3}. 1 b₁ = -1, b₂ = -5 [x]B= = (Simplify your answer.) 3 4, b3 = 15 2 -2, X= 4 2 -1 18

Find the coordinate vector [x]g of x relative to the given basis B = {b₁,b₂, b3}. 1 b₁ = -1, b₂ = -5 [x]B= = (Simplify your answer.) 3 4, b3 = 15 2 -2, X= 4 2 -1 18

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Problem: Finding the Coordinate Vector** **Objective:** Find the coordinate vector \([x]_B\) of \(x\) relative to the given basis \(B = \{b_1, b_2, b_3\}\). **Given:** - \(b_1 = \begin{bmatrix} 1 \\ -1 \\ -5 \end{bmatrix}\), - \(b_2 = \begin{bmatrix} -3 \\ 4 \\ 15 \end{bmatrix}\), - \(b_3 = \begin{bmatrix} 2 \\ -2 \\ 4 \end{bmatrix}\) The vector \(x\) is: - \(x = \begin{bmatrix} 2 \\ -1 \\ 18 \end{bmatrix}\) **Task:** Determine \([x]_B\). Please simplify your answer before finalizing. **Solution:** To find \([x]_B\), express \(x\) as a linear combination of the basis vectors \(b_1, b_2,\) and \(b_3\). This involves solving the following equation for the coefficients \(c_1, c_2,\) and \(c_3\): \[ x = c_1b_1 + c_2b_2 + c_3b_3 \] After solving, present the answer as a vector: \[ [x]_B = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} \] **Note:** Remember to verify the solution by substituting the coefficients back into the equation and confirming the equality with \(x\).
Transcribed Image Text:**Problem: Finding the Coordinate Vector** **Objective:** Find the coordinate vector \([x]_B\) of \(x\) relative to the given basis \(B = \{b_1, b_2, b_3\}\). **Given:** - \(b_1 = \begin{bmatrix} 1 \\ -1 \\ -5 \end{bmatrix}\), - \(b_2 = \begin{bmatrix} -3 \\ 4 \\ 15 \end{bmatrix}\), - \(b_3 = \begin{bmatrix} 2 \\ -2 \\ 4 \end{bmatrix}\) The vector \(x\) is: - \(x = \begin{bmatrix} 2 \\ -1 \\ 18 \end{bmatrix}\) **Task:** Determine \([x]_B\). Please simplify your answer before finalizing. **Solution:** To find \([x]_B\), express \(x\) as a linear combination of the basis vectors \(b_1, b_2,\) and \(b_3\). This involves solving the following equation for the coefficients \(c_1, c_2,\) and \(c_3\): \[ x = c_1b_1 + c_2b_2 + c_3b_3 \] After solving, present the answer as a vector: \[ [x]_B = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} \] **Note:** Remember to verify the solution by substituting the coefficients back into the equation and confirming the equality with \(x\).
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