Ononot — Let V be the vector space of 2 \times 2 matrices. Consider t

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Vectors & Spaceshard

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Let $V$ be the vector space of $2 \times 2$ matrices. Consider the transformation $T(A) = A^T - \text{tr}(A)I$ . What is the matrix representation of $T$ with respect to the basis $E_{11} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, E_{12} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, E_{21} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, E_{22} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ ?