Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 4 - Vector Spaces - 4.4 Exercises - Page 224: 6

Answer

The coordinate vector is; $[x]_{\beta}=\begin{bmatrix}{c_1}\\{c_2}\end{bmatrix} = \begin{bmatrix}-6\\{ 2}\end{bmatrix}$

Work Step by Step

To find the coordinate vector $[x]_{\beta}$ of $x$ relative to the given basis $\beta$ Basis matrix $\beta = \begin{bmatrix}1&5\\{ -2}&{ - 6}\end{bmatrix}$ and ${\bf{x}} = \begin{bmatrix}{ 4}\\0\end{bmatrix}$ finding $[x]_{\beta}$ $\beta {\left[ x \right]_\beta } = x$ $\begin{bmatrix}1&5\\{ -2}&{-6}\end{bmatrix}\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}=\begin{bmatrix}4\\0\end{bmatrix}$ forming augmented matrix and row reducing; $\begin{bmatrix}1&5&4\\{ -2}&{-6}&0\end{bmatrix}\sim\begin{bmatrix}1&0&-6\\0&1&2\end{bmatrix}$ The coordinate vector is; $[x]_{\beta}=\begin{bmatrix}{c_1}\\{c_2}\end{bmatrix} = \begin{bmatrix}-6\\{ 2}\end{bmatrix}$
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