- #1

Mutaja

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## Homework Statement

B = {b

_{1}, b

_{2}, b

_{3}}

and

C = {c

_{1}, c

_{2}, c

_{3}}

are two basis's for R

^{3}where the connection between the basis vectors are given by

b

_{1}= -c

_{1}+ 4c

_{2}, b

_{2}= -c

_{1}+ c

_{2}+ c

_{3}, b

_{3}= c

_{2}- 2c

_{3}

a) decide the transformation matric from basis B to basis C.

A vector x is given in relation to basis B by $$[X]_b=\begin{pmatrix}-3\\ 4\\ 1 \end{pmatrix}$$

b) Decide the coordinates to the vector x in relation to basis C.

## Homework Equations

Transformation matrices between basis B, C and standard.

## The Attempt at a Solution

My problem here is dealing with such a generalized problem. I mean, it shouldn't be any different than if we were dealing with numbers, right? But I can't see how to solve this.

This is my attempt:

The basis B and C contains three vectors each?

$$b_1=\begin{pmatrix}x1\\ x2\\ x3 \end{pmatrix}$$, $$b_2=\begin{pmatrix}x4\\ x5\\ x6 \end{pmatrix}$$, $$b_3=\begin{pmatrix}x7\\ x8\\ x9 \end{pmatrix}$$

$$c_1=\begin{pmatrix}y1\\ y2\\ y3 \end{pmatrix}$$, $$c_2=\begin{pmatrix}y4\\ y5\\ y6 \end{pmatrix}$$, $$c_3=\begin{pmatrix}y7\\ y8\\ y9 \end{pmatrix}$$

To find the transition matrix M

_{B->C}we have to go through the standard basis.

Standard basis: M

_{B->S}

[tex]

\begin{pmatrix}

x1 & x4 & x7 \\ x2 & x5 & x8 \\ x3 & x6 & x9

\end{pmatrix}\quad

[/tex]

M

_{C->S}

[tex]

\begin{pmatrix}

y1 & y4 & y7 \\ y2 & y5 & y8 \\ y3 & y6 & y9

\end{pmatrix}\quad

[/tex]

M

_{B->C}= M

_{S->C}* M

_{B->S}= M

^{-1}

_{C->S}* M

_{B->S}

=(inverse) [tex]

\begin{pmatrix}

x1 & x4 & x7 \\ x2 & x5 & x8 \\ x3 & x6 & x9

\end{pmatrix}\quad

[/tex] * [tex]

\begin{pmatrix}

y1 & y4 & y7 \\ y2 & y5 & y8 \\ y3 & y6 & y9

\end{pmatrix}\quad

[/tex] = M

^{-1}

_{C->S}* M

_{B->S}.

Am I doing the right thing here? I cut it semi-short since I feel like I'm wasting my time doing the wrong thing. I've done it on paper, so if you want me to fill in the last part, just let me know.