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 $C$. $T: P_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{2}$ given by $$T\left(a+b x+c x^{2}\right)=(a-3 c, 2 a+b-2 c)$$ (a) $B=\left\{1, x, x^{2}\right\} ; C=\{(1,0),(0,1)\}$ (b) $B=\left\{1,1+x, 1+x+x^{2}\right\} ; C=\{(1,-1),(2,1)\}$

Determine the matrix representation $[T]_{B}^{C}$ for the given linear transformation $T$ and ordered bases $\bar{B}$ and $C$.
$T: P_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{2}$ given by
$$T\left(a+b x+c x^{2}\right)=(a-3 c, 2 a+b-2 c)$$
(a) $B=\left\{1, x, x^{2}\right\} ; C=\{(1,0),(0,1)\}$
(b) $B=\left\{1,1+x, 1+x+x^{2}\right\} ; C=\{(1,-1),(2,1)\}$

Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves the operations of vector addition and scalar multiplication. This property allows the transformation to be completely determined by its action on a basis of the vector space.

Matrix Representation

The matrix representation of a linear transformation with respect to a pair of ordered bases is the unique matrix that maps the coordinate vector of any input (with respect to the domain basis) to the coordinate vector of its image (with respect to the codomain basis). This matrix simplifies computations and analysis of the linear transformation.

Ordered Bases

An ordered basis is a sequence of linearly independent vectors that spans a vector space. The specific ordering is crucial because it directly affects the coordinate representation of vectors and the corresponding matrix representation of linear transformations.

Coordinate Vectors

Coordinate vectors provide a way to express any vector in a vector space in terms of the components relative to an ordered basis. They are fundamental in converting abstract linear transformations into concrete matrix operations.

Change of Basis

Change of basis involves transforming the representation of vectors (and linear transformations) from one basis to another. This process is important when the given bases for the domain and codomain are not the standard ones, requiring adjustments to the matrix representation of the transformation.

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