Question

Let T : C^2 \to C^2 : T\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x + 2y - iz \\ 2y + iz \\ ix + z - 2z \end{bmatrix}.  FInd [T]B ,  [T]B' and P where

B = \left \{ \begin{bmatrix} 0\\ i\\ 0 \end{bmatrix}, \begin{bmatrix} i\\ 1\\ -1 \end{bmatrix} , \begin{bmatrix} 0\\ 0\\ 2 \end{bmatrix} \right \}, B' = \left \{ \begin{bmatrix} 1\\ -i\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} , \begin{bmatrix} 1\\ i\\ 0 \end{bmatrix} \right \},

[T]_{B'} = P^{-1}[T]_BP​​​​​​​

06 Feb 2024
Answer :
Word Count : 627

To find the matrices \([T]_B\), \([T]_{B'}\), and \(P\), we first need to express the transformation \(T\) in terms of the basis vectors of \(B\) and \(B'\). Then, we can use these expressions to compute the matrices.

Let's denote the basis vectors of \(B\) as \(b_1, b_2, b_3\) and the basis vectors of \(B'\) as \(b_1', b_2', b_3'\).

Given that:
\[ B = \left\{ \begin{bmatrix} 0 \\ i \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix} \right\} \]
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