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Chapter 7: Problem 70

Let \(T: P_{2} \rightarrow P_{2}\) be represented by $$\begin{aligned} T\left(a_{0}+a_{1} x+a_{2} x^{2}\right)=&\left(2 a_{0}+a_{1}-a_{2}\right)+ \left(-a_{1}+2 a_{2}\right) x-a_{2} x^{2} \end{aligned}$$ Find the eigenvalues and eigenvectors of \(T\) relative to the standard basis \(\left\\{1, x, x^{2}\right\\}\).

Short Answer

Expert verified
The eigenvalues of the transformation \(T\) are -1 and 0, with corresponding eigenvectors (1,1,2), (1,1,0), and (0,0,1).

Step by step solution

01

Representation of transformation as a matrix

For a basis \(\left\{1, x, x^{2}\right\}\) of \(P_{2}\), we can represent our transformation \(T\) as a matrix by observing how \(T\) acts on the basis elements. Apply \(T\) to each basis element and express the result as a linear combination of the basis elements. The coefficients then make the columns of our matrix.
02

Calculation of the matrix

Applying \(T\) to the basis elements results in three vectors: \(T(1) = 2\cdot1 + (-1)\cdot x + 0\cdot x^2\), \(T(x) = 2\cdot0 + (-1)\cdot 1 + 2\cdot x^2\), and \(T(x^2) = 2\cdot0 + (-1)\cdot0 - x^2\). Expressing each of these as a column, the matrix of \(T\) is \[\left[ \begin{array}{ccc} 2 & 0 & 0 \ -1 & -1 & 0 \ 0 & 2 & -1 \end{array} \right]\]
03

Compute eigenvalues

The eigenvalues \(\lambda\) of a matrix A are the solutions to the characteristic equation \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix. Here, substituting to our matrix, the characteristic polynomial comes out to be \((2-\lambda)((-1-\lambda)(-1-\lambda) - (2\cdot0))\). Factoring gives \(-\lambda(\lambda+1)^2\), so the eigenvalues are \(\lambda = -1, 0\).
04

Compute eigenvectors

Eigenvectors corresponding to eigenvalue \(\lambda\) are found by solving the system of equations \((A - \lambda I)v = 0\), for \(v\). For \(\lambda=0\), \(A - \lambda I\) is just A. Solving this gives eigenvector (1,1,2). Repeat for \(\lambda=-1\) gives us two more eigenvectors (1,1,0) and (0,0,1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Transformation
A linear transformation is a fundamental concept in linear algebra where functions map vectors from one space to another, while preserving vector addition and scalar multiplication. If you think of a vector as an arrow pointing in a specific direction, a linear transformation will either stretch, shrink, rotate, or reflect this arrow, but it won鈥檛 change the nature of these operations.

In our exercise, we deal with a linear transformation \( T \) acting on polynomials of degree two. Specifically, \( T \) is transforming a polynomial \( a_0 + a_1x + a_2x^2 \) into another polynomial, according to given rules.

The transformation process can be thought of as reshaping a three-dimensional space formed by the coefficients \( a_0, a_1, a_2 \). Linear transformations are crucial because they allow us to study the properties of vector spaces using matrices.
Matrix Representation
Matrix representation is a way to describe linear transformations using matrices, making computations more straightforward. Here, the transformation \( T \) is represented as a matrix by observing how it affects each basis element.

For the basis \( \{1, x, x^2\} \) of \( P_2 \), when \( T \) is applied to these elements, we determine the resultant polynomials as a combination of the basis. These coefficients form the columns of our matrix.

  • Applying \( T \) to 1 gives us the polynomial's constant term.
  • Applying \( T \) to \( x \) affects the linear component.
  • Finally, applying \( T \) to \( x^2 \) primarily transforms the quadratic term.
These operations result in a matrix that provides a clear, numerical representation of the transformation, allowing easier manipulation and analysis.
Characteristic Equation
The characteristic equation is a polynomial equation that helps find the eigenvalues of a matrix, capturing the essence of the transformation in equation form. For a given transformation matrix \( A \), the characteristic equation is derived from the determinant \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix and \( \lambda \) represents the eigenvalues.

In our solution, we formed this equation by substituting the transformation matrix of \( T \) and expanding the determinant to obtain a polynomial. Solving this polynomial gives us values of \( \lambda \) which are the eigenvalues, indicating how scalar factors scale eigenvectors under transformation. Each unique \( \lambda \) represents a distinct way in which the transformation preserves the directional properties of vectors.
Polynomial Basis
A polynomial basis is a set of polynomials that can express any polynomial in a particular polynomial space. In our context, we work with the standard basis \( \{1, x, x^2\} \) of quadratic polynomials \( P_2 \).

This basis allows us to describe any polynomial in \( P_2 \) as a unique combination of these basis polynomials. For example, a polynomial \( a_0 + a_1x + a_2x^2 \) is written as:\( a_0(1) + a_1(x) + a_2(x^2) \).

The choice of basis influences the matrix representation of a linear transformation. By expressing the transformation \( T \) relative to \( \{1, x, x^2\} \), we can systematically compute how the transformation alters each component of every polynomial in \( P_2 \). Choosing the right basis simplifies calculations and helps reveal the structure of the transformation.

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Most popular questions from this chapter

Find an orthogonal matrix \(P\) such that \(P^{T} A P\) diagonalizes \(A .\) Verify that \(P^{T} A P\) gives the proper diagonal form. $$A=\left[\begin{array}{llll} 4 & 2 & 0 & 0 \\ 2 & 4 & 0 & 0 \\ 0 & 0 & 4 & 2 \\ 0 & 0 & 2 & 4 \end{array}\right]$$

Determine whether the matrix is symmetric. $$\left[\begin{array}{rr} 1 & 3 \\ 3 & -1 \end{array}\right]$$

Show that the matrix below is orthogonal for any value of \(\theta\). $$A=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$$

Find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. $$\left[\begin{array}{rrr} 0 & 4 & 4 \\ 4 & 2 & 0 \\ 4 & 0 & -2 \end{array}\right]$$

Determine whether the matrix is symmetric. $$\left[\begin{array}{rrrr} 2 & 0 & 3 & 5 \\ 0 & 11 & 0 & -2 \\ 3 & 0 & 5 & 0 \\ 5 & -2 & 0 & 1 \end{array}\right]$$
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