91Ó°ÊÓ

When working with linear transformations, it's essential to understand the concept of a _basis_. A basis of a vector space is a set of vectors that are linearly independent and span the entire space. This means that any vector in the space can be written uniquely as a linear combination of these basis vectors. The basis vectors provide a coordinate system for the vector space.

In our example, the basis \( \mathcal{B} = \mathcal{C} = \{E_{11}, E_{12}, E_{21}, E_{22}\} \) consists of matrices where each matrix has only one entry as 1 and zeroes elsewhere, representing key positions in a 2x2 matrix. This forms a foundation that allows us to meticulously describe and compute transformations in \( M_{22} \), the space of all 2x2 matrices.

• \( E_{11} \) is \( \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \), indicating the position (1,1).
• \( E_{12} \) is \( \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \), indicating (1,2).
• \( E_{21} \) is \( \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix} \), indicating (2,1).
• \( E_{22} \) is \( \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} \), indicating (2,2).

The choice of basis significantly affects how we perceive and handle transformations within this system.

Matrix transposition is a crucial operation in linear algebra, where the rows of a matrix are swapped with its columns. If you have a matrix \( A \), the transposed matrix, denoted \( A^T \, \) involves flipping \( A \) around its main diagonal.

For example, if \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), then \( A^T = \begin{bmatrix} a & c \ b & d \end{bmatrix} \).

This transformation shifts the elements \( b \) and \( c \, \) affecting the orientation of the matrix while preserving mathematical properties like symmetry. In our problem, the transformation \( T(A) = A^T \) is applied to each of the basis matrices, resulting in:

• \( T(E_{11}) = E_{11}^T = E_{11} \)
• \( T(E_{12}) = E_{12}^T = E_{21} \)
• \( T(E_{21}) = E_{21}^T = E_{12} \)
• \( T(E_{22}) = E_{22}^T = E_{22} \)

These transformations help construct the transformation matrix, which will map any vector in \( V \) to \( W \).

Theorem 6.26 is an essential principle in linear algebra that connects the transformation matrix to the transformation of vectors across different bases. It assures us that applying the transformation to a vector can be faithfully represented by matrix multiplication of the transformation matrix and the coordinate vector.

This theorem is verified by illustrating that applying the linear transformation \( T(A) = A^T \) on a vector \( v \), represented in the basis \( \mathcal{B} \, \) can be achieved equivalently twofold:

• Direct computation: Transpose the vector \( A \) to get \( A^T \).
• Matrix multiplication: Multiply the transformation matrix \( [T]_{C \leftarrow B} \) by the coordinate vector of \( v \).

In our case, for a vector \( v = A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), its coordinate vector is \[ [v]_B = \begin{bmatrix} a \ b \ c \ d \end{bmatrix} \]. \ By calculating \[ [T]_{C \leftarrow B} [v]_B = \begin{bmatrix} a \ c \ b \ d \end{bmatrix} \], \ we corroborate that our matrix transformation provides the correct transposed positions.

Coordinate vectors are methods of representing vectors in terms of a specific basis. They serve as numerical representations that capture how vectors relate to that basis. For a given vector space and its basis, any vector can be expressed as a coordinate vector.

In our exercise, by expressing vectors like \( T(E_{11}) = E_{11} \) and others in terms of the basis \( \mathcal{C} \), \ we convert these expressions into coordinate vectors. Each vector’s impact is recorded through a specific format that aids in constructing the matrix \( [T]_{C \leftarrow B} \).

The coordinate vectors of the transformed matrices form the columns of \( [T]_{C \leftarrow B} \):

• \( T(E_{11}) = E_{11} \rightarrow \begin{bmatrix} 1 \ 0 \ 0 \ 0 \end{bmatrix} \)
• \( T(E_{12}) = E_{21} \rightarrow \begin{bmatrix} 0 \ 0 \ 1 \ 0 \end{bmatrix} \)
• \( T(E_{21}) = E_{12} \rightarrow \begin{bmatrix} 0 \ 1 \ 0 \ 0 \end{bmatrix} \)
• \( T(E_{22}) = E_{22} \rightarrow \begin{bmatrix} 0 \ 0 \ 0 \ 1 \end{bmatrix} \)

This clarity in representation simplifies operations, making transformation analysis intuitive and efficient.