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I'm not sure what to do in the following question.

A linear transformation has matrix

[tex]P = \left[ T \right]_B = \left[ {\begin{array}{*{20}c}

3 & { - 4} \\

1 & { - 1} \\

\end{array}} \right][/tex]

with respect to the standard basis B = {(0,1),(0,1)} for R^2.

a) Give the matrix [tex]Q = \left[ T \right]_{B'} [/tex] with respect to the basis B' = {(3,1),(2,1)}.

b) Determine whether or not the transformation is diagonalizable.

c) Find a general expression for Q^n. Hence deduce the formula for [tex]P^n = \left[ {\begin{array}{*{20}c}

3 & { - 4} \\

1 & { - 1} \\

\end{array}} \right]^n [/tex].

Here is what I've thought about.

a) The format of this question is quite new to me, I haven't seen any questions set out in this way before so I'm just going to have a guess...

[tex]

T\left( {\left[ {\begin{array}{*{20}c}

x \\

y \\

\end{array}} \right]} \right) = \left[ T \right]_b \left[ {\begin{array}{*{20}c}

x \\

y \\

\end{array}} \right] = \left( {3x - 4y,x - y} \right)

[/tex]

Oh wait...is this just an application of transitition matrices? As in, I have [T]_B and I want [T]_B'. I also have the two basis sets, so I can use transition matricies to find [T]_B'?

b) I don't understand this question. I've only done questions on determining whether or not a matrix is diagonalizable. I'm not sure how this relates to the transformation itself. That is, how a matrix of a transformation relates to whether or not the transformation is diagonalizable

c) I don't get the point of this question. How does having Q^n allow me to deduce the formula for P^n? Perhaps if someone tells me what I need to do in part 'a' then it should make this part more clear.

Any help with either of the 3 parts of the question would be good thanks.

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# Homework Help: Linear transformation - matrices

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