/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Prove Theorem 9.1: Let \(f\) and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 7

Prove Theorem 9.1: Let \(f\) and \(g\) be polynomials. For any square matrix \(A\) and scalar \(k\) (i) \(\quad(f+g)(A)=f(A)+g(A)\) (iii) \(\quad(k f)(A)=k f(A)\) (ii) \(\quad(f g)(A)=f(A) g(A)\) (iv) \(f(A) g(A)=g(A) f(A)\)

Short Answer

Expert verified
In summary, we proved Theorem 9.1 as follows: (i) We showed that \((f+g)(A) = f(A) + g(A)\) by comparing the sums of the individual polynomials; (ii) We demonstrated that \((k f)(A) = k f(A)\) by comparing scalar multiplication within the polynomial sum; (iii) We established that \((f g)(A) = f(A) g(A)\) by expanding the polynomial product and comparing the resulting sum; and (iv) We confirmed that \(f(A) g(A) = g(A) f(A)\) by invoking the commutativity of the polynomial product.

Step by step solution

01

Part (i) - Polynomial Addition:

Given that \(f\) and \(g\) are polynomials, we can represent them as: \(f(A) = \sum_{i=0}^n a_i A^i\) and \(g(A) = \sum_{j=0}^m b_j A^j\) Now, let's find \((f+g)(A)\): \((f+g)(A) = \sum_{i=0}^n (a_i + b_i) A^i\) Notice that this sum is just the sum of the two individual polynomials for \(A\): \(f(A) + g(A) = \sum_{i=0}^n a_i A^i + \sum_{j=0}^m b_j A^j\) Since they have the same exponent, we can combine these expressions: \(f(A) + g(A) = \sum_{i=0}^n (a_i + b_i) A^i\) Thus, we have shown that \((f+g)(A) = f(A) + g(A)\).
02

Part (ii) - Scalar Multiplication:

Now let's show that \((k f)(A) = k f(A)\). First, we represent \((k f)(A)\) as: \((k f)(A) = \sum_{i=0}^n k(a_i) A^i\) Now, we find \(k f(A)\): \(k f(A) = k \sum_{i=0}^n a_i A^i = \sum_{i=0}^n k(a_i) A^i\) Thus, we have shown that \((k f)(A) = k f(A)\).
03

Part (iii) - Polynomial Multiplication:

Now let's prove that \((f g)(A) = f(A) g(A)\). Define \(h(A) = f(A) g(A)\). Thus, we have: \(h(A) = (\sum_{i=0}^n a_i A^i)(\sum_{j=0}^m b_j A^j)\) Now, we expand this product: \(h(A) = \sum_{i=0}^n\sum_{j=0}^m a_i b_j A^{i+j}\) Since \(A^{i+j}\) has the same exponent, we can rewrite this as a single sum: \(h(A) = \sum_{k=0}^{n+m}(\sum_{i+j=k} a_i b_j) A^k\) This result shows that \(h(A) = (f g)(A)\), so we have shown that \((f g)(A) = f(A) g(A)\).
04

Part (iv) - Commutativity:

Finally, we will show that \(f(A) g(A) = g(A) f(A)\). From part (iii), we know that \(f(A) g(A) = (f g)(A)\), and similarly, \(g(A) f(A) = (g f)(A)\). Since the product of two polynomials is commutative, i.e., \(f g = gf\), we can write: \(f(A) g(A) = (f g)(A)\), and \(g(A) f(A) = (g f)(A)\) Therefore, \(f(A) g(A) = g(A) f(A)\), which concludes our proof for Theorem 9.1.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Eigenvalues and Eigenvectors of \(2 \times 2\) Matrices Let \(A=\left[\begin{array}{ll}3 & -4 \\ 2 & -6\end{array}\right]\) (a) Find all eigenvalues and corresponding eigenvectors. (b) Find matrices \(P\) and \(D\) such that \(P\) is nonsingular and \(D=P^{-1} A P\) is diagonal.

Show that \(A\) is a scalar matrix \(k I\) if and only if the minimal polynomial of \(A\) is \(m(t)=t-k\)

Suppose \(i \neq 0\) is an cigcnvaluc of the composition \(F \circ G\) of linear operators \(F\) and \(G\). Show that \(\lambda\) is also an eigenvalue of the composition \(G \circ F\). [Hint: Show that \(G(v)\) is an eigenvector of \(G \circ F\).]

Find a matrix \(A\) whose minimal polynomial is \(f(t)=t^{3}-8 t^{2}+5 t+7\). Simply let $A=\left[\begin{array}{lll}0 & 0 & -7 \\ 1 & 0 & -5 \\ 0 & 1 & 8\end{array}\right]\(, the companion matrix of \)f(t)$ [defined in Example 9.12(b)].

Let \(A\) be an \(n\) -square matrix for which \(A^{k}=0\) for some \(k>n .\) Show that \(A^{n}=0\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.