Elementary Linear Algebra 7th. Edition by Ron Larson | 9781133110873 | 1133110878 | Holooly

Elementary Linear Algebra

238 SOLVED PROBLEMS

Question: Appendix.2

Using Mathematical Induction Use mathematical induction to prove the following formula. Sn = 1 + 3 + 5 + 7 + ··· + (2n – 1) = n² ...

Verified Answer:

Mathematical induction consists of two distinct pa...

Question: Appendix.4

Using Mathematical Induction in Linear Algebra If A1, A2, · · ·, An are invertible matrices, then prove the generalization of Theorem 2.9. (A1 A2 A3 · · · An)^-1 = An^-1 · · · A3^-1A2^-1A1^-1 ...

Verified Answer:

1. The formula is valid trivially when n = 1 beca...

Question: Appendix.3

Using Mathematical Induction Use mathematical induction to prove the formula for the sum of the first n squares. Sn = 1² + 2² + 3² + 4² + · · · + n² = n(n + 1)(2n + 1) / 6 ...

Verified Answer:

1. When n = 1, the formula is valid, because [lat...

Question: Appendix.5

Using Proof by Contradiction Prove that √2 is an irrational number. ...

Verified Answer:

Begin by assuming that

\sqrt{2}

is...

Question: Appendix.1

Sum of Odd Integers Use the pattern to propose a formula for the sum of the first n odd integers. 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3+ 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 ...

Verified Answer:

Notice that the sums on the right are equal to the...

Question: Appendix.6

Using Proof by Contradiction A positive integer greater than 1 is a prime when its only positive factors are 1 and itself. Prove that there are infinitely many prime numbers. ...

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Assume there are only finitely many prime numbers,...

Question: Appendix.7

Using Proof by Contradiction in Linear Algebra Let A and B be n × n matrices such that AB is singular. Prove that either A or B is singular. ...

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Assume that neither A or B is singular. Because yo...

Question: Appendix.8

Using a Counterexample Use a counterexample to show that the statement is false. Every odd number is prime. ...

Verified Answer:

Certainly, you can list many odd numbers that are ...

Question: Appendix.9

Using a Counterexample in Linear Algebra Use a counterexample to show that the statement is false. If A and B are square singular matrices of order n, then A + B is a singular matrix of order n. ...

Verified Answer:

Let

A = \begin{bmatrix} 1 & 0 \\ 0 &...

Question: Appendix.10

Using a Counterexample in Linear Algebra Use a counterexample to show that the statement is false. The set of all 2 × 2 matrices of the form [1 b c d] with the standard operations is a vector space. ...

Verified Answer:

To show that the set of matrices of the given form...

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