To find the matrix representation $A$ of the linear transformation that transposes matrices, with respect to the given basis, we need to express how the transpose operation acts on each of the basis matrices. The basis $S$ consists of four $2 \times 2$ matrices:

$S = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right\}.$

Step-by-Step Process:

1. Identify the Basis Matrices: Let these basis matrices be labeled as $B_1, B_2, B_3, B_4$:

   B_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, B_3 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, B_4 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$$

2. Apply the Transpose Operation: The transpose of each matrix will produce: $B_1^T = B_1, \quad B_2^T = B_3, \quad B_3^T = B_2, \quad B_4^T = B_4.$

3. Express the Results in the Given Basis: We express each transposed matrix as a linear combination of the basis matrices $B_1, B_2, B_3, B_4$:

   • $B_1^T = 1 \cdot B_1$
   • $B_2^T = 1 \cdot B_3$
   • $B_3^T = 1 \cdot B_2$
   • $B_4^T = 1 \cdot B_4$

4. Construct the Matrix Representation $A$: The matrix representation of the linear transformation with respect to the basis $S$ is constructed by writing the coefficients from the linear combinations in a matrix. Each column of the matrix represents the result of applying the transpose operation to one of the basis matrices, expressed in the basis $S$.

   Therefore, the matrix $A$ will be:

   1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.$$

This matrix $A$ represents the linear transformation of transposing a $2 \times 2$ matrix in the given basis $S$.

Would you like more details or have any questions?

Related Questions:

1. How can this matrix representation be used in applications?
2. What would the matrix representation look like for a different transformation, such as matrix rotation?
3. How does the choice of basis affect the matrix representation of a transformation?
4. What are some properties of the matrix transpose transformation in higher dimensions?
5. How can we generalize this process for larger matrix spaces?

Tip:

When working with linear transformations, always ensure to apply the transformation directly to the basis vectors to construct the matrix representation efficiently.