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Matrix representation is an essential concept when discussing linear transformations. For any linear transformation from vector space \(V\) to \(W\), the transformation can be represented as a matrix once a basis for each space is established. This matrix allows us to compute the image of vectors under the transformation easily. In the given exercise, we have a specific linear transformation \(T\) defined on \(2 \times 2\) matrices. The task required finding the matrix representation \([T]_{\mathcal{B}}\) with respect to the standard basis \(\mathcal{B}\) for \(M_{22}\). The standard basis for \(2 \times 2\) matrices consists of four matrices \(E_{11}, E_{12}, E_{21}, E_{22}\) each with a single entry of 1 and all other entries 0. The matrix representation reflects how \(T\) acts on each basis element by stacking the results into a matrix where each column corresponds to the image of a basis element.

The standard basis refers to a set that forms an organizational framework for elements in a space. For \(2 \times 2\) matrices, the standard basis is \(\{E_{11}, E_{12}, E_{21}, E_{22}\}\). Each of these matrices has a 1 in one position and 0 in all other positions. For example, \(E_{11}\) is \(\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\), representing a matrix with 1 in the top left corner. This basis allows us to express any \(2 \times 2\) matrix as a unique linear combination of these four matrices. When calculating \(T(E_{11})\), \(T(E_{12})\), \(T(E_{21})\), and \(T(E_{22})\), each result helps form the columns of the matrix \([T]_{\mathcal{B}}\). The standard basis simplifies linear transformation calculations because the results directly relate to familiar and easily manageable matrices.

Matrix computation involves operations like addition, subtraction, and multiplication on matrices, which are key components in finding the results of linear transformations. In this exercise, the primary task involved computing \(A B - B A\) for matrices \(A\) and \(B\) to find \(T(A)\). These computations are fundamental, as they determine how \(T\) affects individual elements of the basis. The calculations for \(T(E_{ij})\) require careful operations that can be broken down into smaller steps:

• Multiply \(E_{ij}\) by matrix \(B\).
• Multiply \(B\) by \(E_{ij}\).
• Calculate their difference \(E_{ij}B - BE_{ij}\).

Each step is crucial to ensure accuracy, as these computed differences comprise the columns of the transformation matrix \([T]_{\mathcal{B}}\). Proper matrix computation requires attention to each element location, ensuring no misplacement or arithmetic errors.

Theorem 6.26 for linear transformations provides that if \(T\) is represented by \([T]_{\mathcal{B}}\) with respect to some basis \(\mathcal{B}\), the image \(T(\mathbf{v})\) for a vector \(\mathbf{v}\) can be computed by applying the matrix \([T]_{\mathcal{B}}\) to the coordinate vector \([\mathbf{v}]_{\mathcal{B}}\). This theorem allows us to check the correctness of matrix representation. In this exercise, verification involved both direct computation of \(T(\mathbf{v})\) using the formula \(AB - BA\) and by multiplying \([T]_{\mathcal{B}}\) with \([\mathbf{v}]_{\mathcal{B}}\). This step validated that both methods yield the same result for \(T(\mathbf{v})\), thus confirming the theorem's correctness. Verification provides confidence in the calculated representation and showcases the utility of matrix representation for understanding and applying transformations efficiently.