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Get Full Access to Linear Algebra - 4 Edition - Chapter 7.3 - Problem 1
Get Full Access to Linear Algebra - 4 Edition - Chapter 7.3 - Problem 1

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# Solved: Label the following statements as true or false. Assume that all vector spaces

ISBN: 9780130084514 53

## Solution for problem 1 Chapter 7.3

Linear Algebra | 4th Edition

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Linear Algebra | 4th Edition

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Problem 1

Label the following statements as true or false. Assume that all vector spaces are finite-dimensional. (a) Every linear operator T has a polynomial p(t) of largest degree for which 7->(T) = TQ. (b) Every linear operator has a unique minimal polynomial. (c) The characteristic polynomial of a linear operator divides the minimal polynomial of that operator. (d) The minimal and the characteristic polynomials of any diagonalizable operator are equal. (e) Let T be a linear operator on an n-dimensional vector space V, p(t) be the minimal polynomial of T, and f(t) be the characteristic polynomial of T. Suppose that f(t) splits. Then f(t) divides \p(t)\n - (f) The minimal polynomial of a linear operator always has the same degree as the characteristic polynomial of the operator. (g) A linear operator is diagonalizable if its minimal polynomial splits. (h) Let T be a linear operator on a vector space V such that V is a T-cyclic subspace of itself. Then the degree of the minimal polynomial of T equals dim(V). (i) Let T be a linear operator on a vector space V such that T has n distinct eigenvalues, where n dim(V). Then the degree of the minimal polynomial of T equals n.

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Math Essentials Notes: WHOLE NUMBERS Place Value, Names for Numbers and Reading Tables  Place Value o 1. 3,450  0 is in the ones place  5 is in the tens place  4 is in the hundreds place  3 is in the thousands place  Standard Form = written in words o 1,083,664,500  One million, Eighty-three million, Six hundred sixty-four thousand, Five hundred o Six Thousand, Four hundred ninety-three  6,493  Expanded Form o 5,672  5,000 + 600 + 70 +2

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