Determine the matrix representation [T]B^C for the given...

Determine the matrix representation $[T]_{B}^{C}$ for the given linear transformation $T$ and ordered bases $\bar{B}$ and $C$. $T: M_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R})$ given by $$T(A)=2 A-A^{T}$$ (a) $B=C=\left\{E_{11}, E_{12}, E_{21}, E_{22}\right\}$ (b) $B=\left\{\left[\begin{array}{ll}-1 & -2 \\ -2 & -3\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 2 & 2\end{array}\right],\left[\begin{array}{ll}0 & -3 \\ 2 & -2\end{array}\right],\left[\begin{array}{ll}0 & 4 \\ 1 & 0\end{array}\right]\right\}$ $C=\left\{E_{11}, E_{12}, E_{21}, E_{22}\right\}$

Determine the matrix representation $[T]_{B}^{C}$ for the given linear transformation $T$ and ordered bases $\bar{B}$ and $C$.
$T: M_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R})$ given by
$$T(A)=2 A-A^{T}$$
(a) $B=C=\left\{E_{11}, E_{12}, E_{21}, E_{22}\right\}$
(b) $B=\left\{\left[\begin{array}{ll}-1 & -2 \\ -2 & -3\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 2 & 2\end{array}\right],\left[\begin{array}{ll}0 & -3 \\ 2 & -2\end{array}\right],\left[\begin{array}{ll}0 & 4 \\ 1 & 0\end{array}\right]\right\}$
$C=\left\{E_{11}, E_{12}, E_{21}, E_{22}\right\}$

Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves the operations of addition and scalar multiplication. This concept is fundamental in linear algebra because it forms the basis for understanding how structures are mapped from one space to another, and it underlies many applications including systems of equations and transformations in geometry.

Matrix Representation of a Linear Transformation

This concept involves expressing a linear transformation as a matrix once bases have been fixed for the domain and codomain. The matrix representation encapsulates the effect of the transformation on any vector by encoding how the basis vectors are mapped, which in turn facilitates computation, analysis, and change of coordinates.

Ordered Basis

An ordered basis for a vector space is a finite set of linearly independent vectors arranged in a specific sequence. The order is crucial because it determines the coordinate representation of vectors relative to that basis and affects the resulting matrix representation of a linear transformation.

Change of Basis

Change of basis refers to the process of converting the representation of vectors and linear transformations from one basis to another. This concept is important because the matrix representation of a linear transformation depends on the chosen ordered bases of the domain and codomain, and understanding this process allows one to navigate between different coordinate systems.

Transpose Operator

The transpose operator is a mapping that flips a matrix over its diagonal, interchanging its row and column indices. This operator has specific algebraic properties and appears frequently in linear algebra, particularly in contexts such as symmetry, orthogonality, and the study of matrix equations.

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