Consider the basis \[\mathcal{B} = \left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \end{pmatrix} \right\}\]of $\mathbb{R}^2$. What is the coordinate vector of \[\begin{pmatrix} 2 \\ 1 \end{pmatrix}\]with respect to $\mathcal{B}$?

Given basis is B=12,25 We have to find the coordinate vector of x¯=21 Let the coordinate vector of…

Answered: Consider the basis 8-{(!). (3)} 1 2 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

Consider the basis

\[
\mathcal{B} = \left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \end{pmatrix} \right\}
\]

of $\mathbb{R}^2$. What is the coordinate vector of

\[
\begin{pmatrix} 2 \\ 1 \end{pmatrix}
\]

with respect to $\mathcal{B}$?

Expert Solution

Step 1

Given basis is $B = \{(\begin{array}{l} 1 \\ 2 \end{array}), (\begin{array}{l} 2 \\ 5 \end{array})\}$

We have to find the coordinate vector of $\bar{x} = (\begin{array}{l} 2 \\ 1 \end{array})$

Let the coordinate vector of $\bar{x} = (\begin{array}{l} 2 \\ 1 \end{array})$ is ${[\bar{x}]}_{B} = (\begin{array}{l} x_{1} \\ x_{2} \end{array})$

Then $x_{1} b_{1} + x_{2} b_{2} = \bar{x}$

Hence,

$\begin{matrix} x_{1} b_{1} + x_{2} b_{2} = \bar{x} \\ x_{1} (\begin{array}{l} 1 \\ 2 \end{array}) + x_{2} (\begin{array}{l} 2 \\ 5 \end{array}) = (\begin{array}{l} 2 \\ 1 \end{array}) \\ (\begin{array}{l} x_{1} \\ 2 x_{1} \end{array}) + (\begin{array}{l} 2 x_{2} \\ 5 x_{2} \end{array}) = (\begin{matrix} 2 \\ 1 \end{matrix}) \\ (\begin{array}{l} x_{1} + 2 x_{2} \\ 2 x_{1} + 5 x_{2} \end{array}) = (\begin{array}{l} 2 \\ 1 \end{array}) \end{matrix}$

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