- #1
Mutaja
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Homework Statement
B = {b1, b2, b3}
and
C = {c1, c2, c3}
are two basis's for R3 where the connection between the basis vectors are given by
b1 = -c1 + 4c2, b2 = -c1 + c2 + c3, b3 = c2 - 2c3
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 MB->C we have to go through the standard basis.
Standard basis: MB->S
[tex]
\begin{pmatrix}
x1 & x4 & x7 \\ x2 & x5 & x8 \\ x3 & x6 & x9
\end{pmatrix}\quad
[/tex]
MC->S
[tex]
\begin{pmatrix}
y1 & y4 & y7 \\ y2 & y5 & y8 \\ y3 & y6 & y9
\end{pmatrix}\quad
[/tex]
MB->C = MS->C * MB->S = M-1C->S * MB->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-1C->S * MB->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.