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Chapter 3: Problem 24

Let \(T: P_{3} \rightarrow P_{2}\) be defined by \(T(p(x))=\) \(p^{\prime}(2 x+1)\), where \(p^{\prime}(x)=D(p(x))\), and let \(B\) \(=\left(x^{3}, x^{2}, x, 1\right)\) and \(B^{\prime}=\left(x^{2}, x, 1\right) .\) a. Find the matrix \(A\). b. Use \(A\) to compute \(T\left(4 x^{3}-5 x^{2}+4 x-7\right)\).

Short Answer

Expert verified
Matrix A transforms basis B to B'. Use A to find transformation of given polynomial in terms of B'.

Step by step solution

01

Differentiate the Polynomial

Given a polynomial \( p(x) = a_3x^3 + a_2x^2 + a_1x + a_0 \), first find the derivative \( p'(x) = 3a_3x^2 + 2a_2x + a_1 \). This will be used to find the image under the transformation \( T \).
02

Substitute and Simplify

Substitute \( 2x + 1 \) into the derivative found in Step 1: \( p'(2x+1) = 3a_3(2x+1)^2 + 2a_2(2x+1) + a_1 \). Expand this expression and simplify. This gives you the polynomial in \( P_2 \) resulting from the transformation.
03

Align with Basis B'

Express the resulting polynomial in terms of the basis \( B' = (x^2, x, 1) \). Determine the coefficients for \( x^2, x, \) and the constant term.
04

Construct the Matrix A

Repeat Steps 1 to 3 for the basis elements \( x^3, x^2, x, 1 \) of \( B \). Each result gives a column of the matrix \( A \). Combine these to form the complete matrix \( A \).
05

Verify Matrix A

Ensure the matrix \( A \) correctly transforms each basis element of \( B \) into its image in terms of \( B' \). This can be done by checking that multiplying \( A \) by the coordinates of each \( B \) basis element yields the correct coordinates in \( B' \).
06

Apply Matrix A to a Polynomial

Express the polynomial \( 4x^3 - 5x^2 + 4x - 7 \) in terms of the basis \( B \). This gives you a vector that can be multiplied by \( A \).
07

Calculate the Image Using A

Multiply the matrix \( A \) by the coordinate vector from Step 6 to find \( T(4x^3 - 5x^2 + 4x - 7) \). This multiplication will yield the coordinates in terms of the basis \( B' \).
08

Interpret Result in Basis B'

Convert the resulting vector from Step 7 back into a polynomial expressed in terms of the basis \( B' \) to find \( T(4x^3 - 5x^2 + 4x - 7) \).

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Key Concepts

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

Polynomial Derivatives
Polynomials are mathematical expressions involving variables and coefficients, with operations like addition, subtraction, multiplication, and non-negative integer exponents of variables. A key concept in calculus related to polynomials is differentiation.
Differentiation involves finding the derivative of a polynomial, which represents the rate at which the polynomial's value changes concerning its variable. For a polynomial of the form:

  • \( p(x) = a_3x^3 + a_2x^2 + a_1x + a_0 \), the derivative is calculated as follows:


  • The term \( a_3x^3 \) turns into \( 3a_3x^2 \).

  • The term \( a_2x^2 \) turns into \( 2a_2x \).

  • The term \( a_1x \) turns into \( a_1 \), while

  • the constant \( a_0 \) disappears, as its rate of change is zero.This results in the derivative polynomial:
\( p'(x) = 3a_3x^2 + 2a_2x + a_1 \).
The derivative is hugely important in transformations, especially in understanding how changes in variables affect the entire polynomial.
Linear Transformations
Linear transformations are processes that map vectors (or, in this case, polynomials) from one space to another while preserving the operations of addition and scalar multiplication. When we say a transformation is linear, it satisfies two main properties:
  • Additivity: \( T(u + v) = T(u) + T(v) \) for any vectors \( u \) and \( v \).

  • Scalar Multiplication: \( T(cu) = cT(u) \) for any vector \( u \) and scalar \( c \).

In the context of the exercise, the function \( T(p(x)) = p'(2x + 1) \) is a linear transformation from polynomials of degree 3 (\( P_3 \)) to polynomials of degree 2 (\( P_2 \)). This transformation reduces the polynomial's degree by converting it into its derivative, then further modifies it by substituting \( 2x + 1 \) into the derivative.
The steps involve first differentiating the polynomial and then substituting, ensuring that the transformation is linear by verifying it retains these two fundamental properties. The transformation matrix \( A \) helps in systematically applying this transformation through matrix multiplication, making it easy to calculate outputs for different polynomial inputs.
Basis Change
In linear algebra, a basis is essentially a set of vectors from which you can construct any other vector in the space by linear combination. Changing the basis involves expressing vectors in terms of a different set of vectors, effectively transforming the vector space's foundation.
Consider the polynomial spaces involved in the exercise:
  • \( B = (x^3, x^2, x, 1) \)
  • \( B' = (x^2, x, 1) \)

These represent the original and transformed basis, respectively. A basis change is needed when transforming the polynomial from its original form to its derivative form, which is a dimension reduction from \( P_3 \) to \( P_2 \).
When performing the transformation \( T \), the coefficients of a polynomial in terms of the original basis \( B \) are captured in a vector, which is multiplied by the transformation matrix \( A \).
This matrix is constructed by applying the transformation to each basis element in \( B \) and expressing the outcome in terms of the new basis \( B' \). The output is a set of coefficients aligned not with the original basis but with the transformed one, \( B' \), reflecting the reduced degree and structural changes of the polynomial.

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Most popular questions from this chapter

In Exercises 9-16, determine whether the given set is closed under the usual operations of addition and scalar multiplication, and is a (real) vector space. $$ \text { The set }\\{0\\} \text { consisting only of the number } 0 \text {. } $$

Let \(F\) be the vector space of funciions mapping \(\mathbb{R}\) into \(\mathbb{R}\). Show that a. \(\operatorname{sp}\left(\sin ^{2} x, \cos ^{2} x\right)\) contains all constant functions, b. \(\operatorname{sp}\left(\sin ^{2} x, \cos ^{2} x\right)\) contains the function \(\cos 2 x\), c. \(\operatorname{sp}\left(7, \sin ^{2} 2 x\right)\) contains the function \(8 \cos 4 x\).

Let \(V\) and \(V^{\prime}\) be vector spaces. Mark each of the following True or False. a. A linear transformation of vector spaces preserves the vector-space operations. b. Every function mapping \(V\) into \(V^{\prime}\) relates the algebraic structure of \(V\) to that of \(V^{\prime}\). c. A linear transformation \(T: V \rightarrow V^{\prime}\) carries the zero vector of \(V\) into the zero vector of \(V^{\prime}\). d. A linear transformation \(T: V \rightarrow V^{\prime}\) carries a pair \(v,-v\) in \(V\) into a pair \(v^{\prime},-v^{\prime}\) in \(V^{\prime}\). e. For every vector \(\mathrm{b}^{\prime}\) in \(V^{\prime}\), the function \(T_{\mathrm{y}}: V \rightarrow V^{\prime}\) defined by \(T_{v}(v)=\mathrm{b}^{\prime}\) for all \(\mathbf{v}\) in \(V\) is a linear transformation. f. The function \(T_{\sigma^{:}}: V \rightarrow V^{\prime}\) defined by \(T_{v}(v)=0^{\prime}\), the zero vector of \(V^{\prime}\), for all \(?\) in \(V\) is a linear transformation. g. The vector space \(P_{10}\) of polynomials of degree \(\leq 10\) is isomorphic to \(\mathbb{R}^{10}\). h. There is exactly one isomorphism \(T: P_{10} \rightarrow \mathrm{R}^{4 !}\). i. Let \(V\) and \(V^{\prime}\) be vector spaces of dimensions \(n\) and \(m\), respectively. A linear transformation \(T: V \rightarrow V^{\prime}\) is invertible if and only if \(m=n\). j. If \(T\) in part (i) is an invertible transformation, then \(m=n\).

In Exercises 1-8, decide whether or not the given set, together with the indicated operations of addition and scalar multiplication, is a (real) vector space. The set of all \(2 \times 2\) matrices, with the usual addition but with scalar multiplication defined by \(r A=0\), the \(2 \times 2\) zero matrix.

In Exercises 9-16, determine whether the given set is closed under the usual operations of addition and scalar multiplication, and is a (real) vector space. The set \(P_{n}\) of all polynomials in \(x\), with real coefficients and of degree less than or equal to \(n\), together with the zero polyncmial.
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