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

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

Key Concepts

Ordered Basis

An ordered basis is a sequence of vectors in a vector space that are linearly independent and span the space. The order is significant because it determines the position of each coordinate in the coordinate vector, which in turn affects the matrix representation of any linear transformation applied to the vector space.

Derivative Operator on Polynomial Spaces

The derivative operator, defined on polynomial spaces, is inherently a linear transformation because it satisfies the linearity properties of differentiation. In this problem, the operator maps a polynomial in P3(?) to its derivative in P2(?), demonstrating how calculus concepts can be analyzed using linear algebra.

Matrix Representation of a Linear Transformation

The matrix representation of a linear transformation provides a concrete method to encode the transformation as a matrix, relative to chosen ordered bases for the domain and codomain. Each column of the matrix corresponds to the coordinates of the image of a basis vector from the domain expressed in the codomain basis.

Linear Transformation

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In the context of the problem, the transformation takes one polynomial from P3(?) to another in P2(?) while maintaining the structure inherent to these spaces.

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