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Chapter 7: Problem 7

Let \(B=\left(1, x, x^{2}\right)\) be an ordered basis for $$ V=\left\\{a_{0}+a_{1} x+a_{2} x^{2} \mid a_{0}, a_{1}, a_{2} \in \mathbb{R}\right\\} $$ and let \(B^{\prime}=\left(x^{3}, x^{2}, x, 1\right)\) be an ordered basis for $$ W=\left\\{a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3} \mid a_{0}, a_{1}, a_{2}, a_{3} \in \mathbb{R}\right\\} $$ Find the matrix for the operator \(\mathcal{I}: V \rightarrow W\) defined by $$ \mathcal{I} p(x)=\int_{1}^{x} p(t) d t $$ relative to these bases.

Short Answer

Expert verified
The matrix is \[ \begin{bmatrix} 0 & 0 & \frac{1}{3} \\ 0 & \frac{1}{2} & 0 \\ 1 & 0 & 0 \\ -1 & -\frac{1}{2} & -\frac{1}{3} \end{bmatrix} \].

Step by step solution

01

Understand the Transformation

The operator \( \mathcal{I} \) integrates a polynomial \( p(x) = a_0 + a_1 x + a_2 x^2 \) from 1 to \( x \). The goal is to determine how this operation transforms the basis vectors of \( V \) when expressed in the basis \( B' \) of \( W \).
02

Integrate Basis Elements of V

We integrate each basis element of \( V \) over the interval from 1 to \( x \) to get their images in \( W \).1. \( \mathcal{I}(1) = \int_{1}^{x} 1 \, dt = x - 1 \)2. \( \mathcal{I}(x) = \int_{1}^{x} t \, dt = \frac{x^2}{2} - \frac{1}{2} \)3. \( \mathcal{I}(x^2) = \int_{1}^{x} t^2 \, dt = \frac{x^3}{3} - \frac{1}{3} \)
03

Express Results in Basis B'

Express each integrated form as a linear combination of \( B' = (x^3, x^2, x, 1) \).1. \( x - 1 = 0 \cdot x^3 + 0 \cdot x^2 + 1 \cdot x - 1 \cdot 1 \)2. \( \frac{x^2}{2} - \frac{1}{2} = 0 \cdot x^3 + \frac{1}{2} \cdot x^2 + 0 \cdot x - \frac{1}{2} \cdot 1 \)3. \( \frac{x^3}{3} - \frac{1}{3} = \frac{1}{3} \cdot x^3 + 0 \cdot x^2 + 0 \cdot x - \frac{1}{3} \cdot 1 \)
04

Form the Transformation Matrix

The coordinates of these expressions in basis \( B' \) form the columns of the transformation matrix.- For \( \mathcal{I}(1) \), the coordinate vector is \( \begin{bmatrix} 0 & 0 & 1 & -1 \end{bmatrix}^T \)- For \( \mathcal{I}(x) \), the coordinate vector is \( \begin{bmatrix} 0 & \frac{1}{2} & 0 & -\frac{1}{2} \end{bmatrix}^T \)- For \( \mathcal{I}(x^2) \), the coordinate vector is \( \begin{bmatrix} \frac{1}{3} & 0 & 0 & -\frac{1}{3} \end{bmatrix}^T \)Thus, the transformation matrix is \[ \begin{bmatrix} 0 & 0 & \frac{1}{3} \ 0 & \frac{1}{2} & 0 \ 1 & 0 & 0 \ -1 & -\frac{1}{2} & -\frac{1}{3} \end{bmatrix} \]

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Key Concepts

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

Basis Transformation
A basis transformation is a method of changing the basis in which a vector space is expressed. It helps us understand how different basis sets relate to each other. In our exercise, we have two different bases: \(B = (1, x, x^2)\) for the vector space \(V\) and \(B' = (x^3, x^2, x, 1)\) for the vector space \(W\). When we perform a basis transformation, we describe vectors from one basis in terms of another basis.
  • We look at how each element of one basis is expressed as a combination of the elements of another basis.
  • This is fundamental in understanding how linear transformations act under different bases.
In this context, the transformation matrix gives us a systematic way to convert between these bases. This conversion is essential for ensuring the transformation \(\mathcal{I}\) is accurately represented.
Integral Operator
An integral operator is a mathematical function that maps one function to another through integration. It's like a tool that changes a function into a new form by calculating the area under its curve within specific limits.
In our exercise, the integral operator \(\mathcal{I}\) transforms a polynomial \(p(x) = a_0 + a_1 x + a_2 x^2\) into another polynomial by integrating from 1 to \(x\). Here's how it works:
  • \(\mathcal{I}(1) = \int_{1}^{x} 1 \, dt = x - 1\)
  • \(\mathcal{I}(x) = \int_{1}^{x} t \, dt = \frac{x^2}{2} - \frac{1}{2}\)
  • \(\mathcal{I}(x^2) = \int_{1}^{x} t^2 \, dt = \frac{x^3}{3} - \frac{1}{3}\)
What happens here is that each basis element from \(V\) is integrated, resulting in polynomial expressions in \(W\). This process showcases how integration acts purely on the functionality of the polynomials rather than on individual coefficients.
Polynomial Basis
A polynomial basis is a set of polynomial functions that span a vector space of polynomials. For example, the basis \(B = (1, x, x^2)\) forms a foundation for all polynomials of degree 2 or less.
  • Each polynomial in this vector space can be expressed as a linear combination of the basis polynomials.
  • Using a polynomial basis helps simplify the expression and manipulation of polynomial functions.
In the exercise, changing between polynomial bases is essential as the action of the integral operator changes the degree of polynomials. Thus, understanding bases like \(B\) for \(V\) and \(B'\) for \(W\) is critical because they dictate how projections and combinations are formed when describing the transformations.
Matrix Representation
Matrix representation is a way of expressing a linear transformation using a matrix. The columns of this matrix correspond to how basis vectors of the domain are transformed into the codomain.
  • The transformation matrix provides a convenient framework to perform and visualize linear transformations across different basis sets.
  • In the given problem, the transformation matrix is constructed by converting results from the integral operator into coordinates relative to the basis \(B'\).
For the integral transformation \(\mathcal{I}\), the matrix representation is:\[\begin{bmatrix}0 & 0 & \frac{1}{3} \0 & \frac{1}{2} & 0 \1 & 0 & 0 \-1 & -\frac{1}{2} & -\frac{1}{3}\end{bmatrix}\]This matrix shows how each basis vector from \(V\) after being transformed can be expressed in terms of \(B'\). It acts as a bridge between the algebraic operation of integration and the geometric notion of basis transformations.

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Most popular questions from this chapter

An example of an operation which is not associative is the cross product. (a) Give a simple example of three vectors from 3-space \(u, v, w\) such that \(u \times(v \times w) \neq(u \times v) \times w\) (b) We saw in Chapter 1 that the operator \(B=u \times(\) cross product with a vector) is a linear operator. It can therefore be written as a matrix (given an ordered basis such as the standard basis). How is it that composing such linear operators is non-associative even though matrix multiplication is associative?

A matrix \(A\) is called anti-symmetric (or skew-symmetric) if \(A^{T}=-A\). Show that for every \(n \times n\) matrix \(M,\) we can write \(M=A+S\) where \(A\) is an anti-symmetric matrix and \(S\) is a symmetric matrix. Hint: What kind of matrix is \(M+M^{T}\) ? How about \(M-M^{T}\) ?

Show that if the range (remember that the range of a function is the set of all its outputs, not the codomain) of a \(3 \times 3\) matrix \(M\) (viewed as a function \(\left.\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\right)\) is a plane then one of the columns is a sum of multiples of the other columns. Show that this relationship is preserved under EROs. Show, further, that the solutions to \(M x=0\) describe this relationship between the columns.

Let's prove the theorem \((M N)^{T}=N^{T} M^{T}\). Note: the following is a common technique for proving matrix identities. (a) Let \(M=\left(m_{j}^{i}\right)\) and let \(N=\left(n_{j}^{i}\right) .\) Write out a few of the entries of each matrix in the form given at the beginning of section 7.3 . (b) Multiply out \(M N\) and write out a few of its entries in the same form as in part (a). In terms of the entries of \(M\) and the entries of \(N,\) what is the entry in row \(i\) and column \(j\) of \(M N ?\) (c) Take the transpose \((M N)^{T}\) and write out a few of its entries in the same form as in part (a). In terms of the entries of \(M\) and the entries of \(N\), what is the entry in row \(i\) and column \(j\) of \((M N)^{T}\) ? (d) Take the transposes \(N^{T}\) and \(M^{T}\) and write out a few of their entries in the same form as in part (a). (e) Multiply out \(N^{T} M^{T}\) and write out a few of its entries in the same form as in part a. In terms of the entries of \(M\) and the entries of \(N,\) what is the entry in row \(i\) and column \(j\) of \(N^{T} M^{T} ?\) (f) Show that the answers you got in parts (c) and (e) are the same.

Above, we showed that left multiplication by an \(r \times s\) matrix \(N\) was a linear transformation \(M_{k}^{s} \stackrel{N}{\longrightarrow} M_{k}^{r} .\) Show that right multiplication by a \(k \times m\) matrix \(R\) is a linear transformation \(M_{k}^{s} \stackrel{R}{\longrightarrow} M_{m}^{s} .\) In other words, show that right matrix multiplication obeys linearity.
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