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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q12E (page 371)

Consider a polynomial $f (位)$ with real coefficients. Show that if a complex number $位_{0}$ is a root of $f (位)$ , then so is its complex conjugate, ${\overset{鈫�}{位}}_{0}$ .

Short Answer

Expert verified

The complex conjugate of $位_{0} is f (\overset{炉}{位_{0}}) = 0$ .

Step by step solution

01

Define the complex number

A complex number is basically a combination of real number and symbolic number. The complex number is in the form of a + ib.

02

Step 2:Find the complex number λ0 is a root of f (λ)

Let $f (位) = a_{n} 位^{n} + a_{n} 位^{n - 1} + . . . + a_{1} 位 + a_{0}$

Where all the coefficients are real, and $位_{0} be a root of f$

Then,

$\begin{aligned} f (\overset{炉}{位_{0}}) = a_{n} \overset{炉}{位_{0}^{n}} + a_{n 鈭� 1} {\overset{炉}{位}}_{00}^{n 鈭� 1} + 鈥� + a_{1} \overset{炉}{位_{0}} + a_{0} \\ = a_{n} \overset{炉}{位_{0}^{n}} + a_{n 鈭� 1} \overset{炉}{位_{0}^{n 鈭� 1}} + 鈥� + a_{1} \overset{炉}{位_{0}} + a_{0} \\ = \overset{炉}{a_{n} 位_{00}^{n} + a_{n 鈭� 1} 位_{0}^{n 鈭� 1} + 鈥� + a_{1} 位_{0} + a_{0}} \\ = \overset{炉}{0} \\ = 0 \end{aligned}$

So, $\overset{炉}{位_{0}} is also a root of f$ .

The solution is the complex conjugate of $位_{0}$ is $f (\overset{炉}{位_{0}}) = 0$ .

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Most popular questions from this chapter

Find all the polynomials of degree $鈮� 2$ [a polynomial of the form $f (t) = a + b t + c t^{2}$ ] whose graph goes through the points (1,3) and (2,6) , such thatrole="math" localid="1659541039431" $f' (1) = 1$ [where $f' (1)$ denotes the derivative].

For an arbitrary positive integer n, give a $2 n 脳 2 n$ matrix A without real eigenvalues.

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

$(\begin{matrix} 0 & - 1 \\ 1 & 2 \end{matrix})$

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

$[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

Find all $2 脳 2$ matrices for which ${\overset{鈬赌}{e}}_{1}$ is an eigenvector.

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