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 $C$. $T: M_{2}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R})$ given by $$T\left(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\right)=(a-d)+3 b x^{2}+(c-a) x^{3}$$ (a) $B=\left\{E_{11}, E_{12}, E_{21}, E_{22}\right\} ; C=\left\{1, x, x^{2}, x^{3}\right\}$ (b) $B=\left\{E_{21}, E_{11}, E_{22}, E_{12}\right\} ; C=\left\{x, 1, x^{3}, x^{2}\right\}$

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

Key Concepts

Coordinate Vectors

Coordinate vectors are the representations of vectors with respect to a particular ordered basis. They provide a way to uniquely express any vector as a linear combination of the basis vectors, and these coordinates are used as the entries in the columns of the matrix representation of a linear transformation.

Ordered Bases

An ordered basis is a sequence of vectors that not only spans a vector space but also establishes a specific, unambiguous coordinate system. The order of the basis vectors is essential because it directly affects the structure and entries of the matrix representation of a linear transformation.

Change of Basis

Change of basis refers to the process of converting the coordinates of vectors from one basis to another. This is important in the context of matrix representations because different choices of bases for the domain and codomain yield different matrix forms for the same linear transformation, affecting how operations like composition and inversion are performed.

Linear Transformation

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. Understanding linear transformations is crucial because they allow us to study the structure of vector spaces and to represent functions as matrices, which simplifies many computations in linear algebra.

Matrix Representation of a Linear Transformation

The matrix representation of a linear transformation is a way to encode the transformation as a matrix once bases for the domain and codomain are fixed. This representation enables the computation of images of vectors through matrix multiplication, effectively translating abstract linear operations into concrete numerical calculations.

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