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

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

Key Concepts

Coordinate Vector Expansion

When forming a matrix representation, each image of the basis vector from the domain under the transformation is expressed as a linear combination of the codomain's basis vectors. The coefficients obtained in these expansions serve as the columns of the transformation matrix, linking the abstract transformation to concrete computations.

Ordered Basis

An ordered basis is a specific sequence of linearly independent vectors that spans a vector space. The order of the basis is significant since it determines a unique coordinate representation for every vector in the space, which is crucial for forming the matrix representation of a linear transformation.

Matrix Representation of a Linear Transformation

The matrix representation of a linear transformation with respect to given ordered bases is a matrix whose columns are the coordinate representations of the images of the basis vectors of the domain expressed in the codomain’s basis. This representation makes it easier to compute the transformation on any vector by matrix multiplication.

Polynomial Vector Spaces

Polynomial vector spaces, such as P?(?) and P?(?), consist of all polynomials of a certain maximum degree with real coefficients. These spaces are important examples in linear algebra because they exhibit the properties of vector spaces and allow exploration of concepts like basis and dimension.

Linear Transformation

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. This concept is fundamental because it allows one to study the structure of the vector spaces and how elements are mapped between them in a way that respects the linear operations.

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