Question

Suppose \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{2}\) is a linear transformation. If \(B=\left\\{1, x, x^{2}\right\\}\) and \(D=\\{(1,1),(0,1)\\},\) find the action of \(T\) given: a. \(M_{D B}(T)=\left[\begin{array}{rrr}1 & 2 & -1 \\ -1 & 0 & 1\end{array}\right]\) b. \(M_{D B}(T)=\left[\begin{array}{rrr}2 & 1 & 3 \\ -1 & 0 & -2\end{array}\right]\)

Short answer

In Case a: \(T(1)=(1,-1)\), \(T(x)=(2,0)\), \(T(x^2)=(-1,1)\); in Case b: \(T(1)=(2,-1)\), \(T(x)=(1,0)\), \(T(x^2)=(3,-2)\).

Step-by-step solution

Understand the Problem

We are given two situations where a linear transformation \(T: \mathbf{P}_{2} \rightarrow \mathbb{R}^{2}\) is described by different matrix representations \(M_{D B}(T)\). Our task is to find the action of \(T\) on the basis elements \(1, x, x^2\) of \(\mathbf{P}_{2}\).

Matrix Representation Insights

The matrix \(M_{D B}(T)\) represents how the linear transformation \(T\) acts on the basis \(B\) of \(\mathbf{P}_{2}\) when the output is expressed in terms of the basis \(D\) of \(\mathbb{R}^{2}\). Each column of the matrix corresponds to the image of a basis element \(B\) under \(T\).

Calculate the Action in Case a

For \(M_{D B}(T)=\begin{bmatrix}1 & 2 & -1 \ -1 & 0 & 1\end{bmatrix}\):- The first column \(\begin{bmatrix}1\-1\end{bmatrix}\) corresponds to \(T(1) = (1,-1)\).- The second column \(\begin{bmatrix}2\0\end{bmatrix}\) corresponds to \(T(x) = (2,0)\).- The third column \(\begin{bmatrix}-1\1\end{bmatrix}\) corresponds to \(T(x^2) = (-1,1)\).

Calculate the Action in Case b

For \(M_{D B}(T)=\begin{bmatrix}2 & 1 & 3 \ -1 & 0 & -2\end{bmatrix}\):- The first column \(\begin{bmatrix}2\-1\end{bmatrix}\) corresponds to \(T(1) = (2,-1)\).- The second column \(\begin{bmatrix}1\0\end{bmatrix}\) corresponds to \(T(x) = (1,0)\).- The third column \(\begin{bmatrix}3\-2\end{bmatrix}\) corresponds to \(T(x^2) = (3,-2)\).

Key concepts

Matrix Representation in Linear Transformations

A matrix representation of a linear transformation provides a structured way to understand how the transformation acts on vectors or basis elements. In our example, we have a linear transformation \( T: \mathbf{P}_{2} \to \mathbb{R}^2 \). This means we are mapping from a space of polynomials up to degree 2, \( \mathbf{P}_{2} \), to \( \mathbb{R}^2 \), a 2-dimensional vector space. The representation matrix \( M_{D B}(T) \) helps us easily determine the output of our transformation when applied to the basis elements.

• The matrix \( M_{D B}(T) \) is rectangular and has as many rows as there are dimensions in \( \mathbb{R}^2 \) and as many columns as there are elements in the basis of \( \mathbf{P}_{2} \).
• Each column in the matrix represents the image of a basis element from \( \mathbf{P}_{2} \) under the transformation \( T \) given in terms of the basis \( D \) of \( \mathbb{R}^2 \).

For example, in scenario (a), the first column \( \begin{bmatrix} 1 \ -1 \end{bmatrix} \) tells us that \( T(1) = (1, -1) \). Breaking down the entries allows a detailed understanding of transformations.

Understanding Basis Elements

Basis elements are the building blocks of vector spaces. A basis is a set of vectors that are linearly independent and span the entire vector space. This means any element in the space can be expressed as a linear combination of these basis elements. For \( \mathbf{P}_{2} \), the basis \( B = \{1, x, x^2 \} \) implies each polynomial of degree at most 2 can be formed as \( a + bx + cx^2 \).

• Each polynomial of degree 2 can thus be uniquely represented by a combination of these basis elements.
• The transformation \( T \) acts on each of these basis elements, and the images are recorded in the columns of its matrix representation \( M_{D B}(T) \).

In our problem, knowing how \( 1, x, x^2 \) are transformed helps us understand the entire action of \( T \), since any other polynomial combination can also be computed using these results.

Exploring Polynomial Vector Spaces

Polynomial vector spaces are fascinating structures in linear algebra. They consist of polynomials of a specific degree, forming a vector space through polynomial addition and scalar multiplication. In \( \mathbf{P}_{2} \), we consider polynomials up to degree 2, such as \( a + bx + cx^2 \).

• \( \mathbf{P}_{2} \) is a 3-dimensional vector space, reflected by the three basis elements \( \{1, x, x^2\} \).
• This space allows linear transformations to map these polynomials to other vector spaces, like \( \mathbb{R}^2 \) in our question.

Understanding polynomial vector spaces gives insight into how complex transformations like \( T \) are formulated and represented, allowing detailed computations and predictions about their behavior. It's a broad and intriguing field, underscoring many algebraic concepts and practical applications.