Word Count : 541

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Linear map $T:\mathbb C^3\to\mathbb C^3$ $$ T\begin{pmatrix}x\\y\\z\end{pmatrix} =\begin{pmatrix}x+2y-i z\\[4pt]2y+i z\\[4pt] i x - z\end{pmatrix}. $$ Bases $$ B=\Big\{b_1=\begin{pmatrix}0\\i\\0\end{pmatrix},\; b_2=\begin{pmatrix}i\\1\\-1\end{pmatrix},\; b_3=\begin{pmatrix}0\\0\\2\end{pmatrix}\Big\},\qquad B'=\Big\{b'_1=\begin{pmatrix}1\\-i\\1\end{pmatrix},\; b'_2=\begin{pmatrix}0\\0\\1\end{pmatrix},\; b'_3=\begin{pmatrix}1\\i\\0\end{pmatrix}\Big\}. $$ Form the matrices whose columns are the basis vectors (these convert coordinates in the basis to standard coordinates): $$ S=[\,b_1\ b_2\ b_3\,]= \begin{pmatrix} 0 & i & 0\\[4pt] i & 1 & 0\\[4pt] 0 & -1 & 2 \end{pmatrix},\qquad S'=[\,b'_1\ b'_2\ b'_3\,]= \begin{pmatrix} 1 & 0 & 1\\[4pt] - i & 0 & i\\[4pt] 1 & 1 & 0 \end{pmatrix}. $$ --- ## 1) Compute $[T]_B$ For each $b_j\in B$ compute $T(b_j)$ (in standard coordinates) and then express $T(b_j)$ as a linear combination of _______ ____ _________ ________ _______.
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