HW4Math211S11

HW4Math211S11 - For f x = x n f x = nx n-1(You may use the...

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Math 211: Linear Algebra Homework 4 Due March 4 Part I Read Sections 1.3 and 1.4 in Leon, and do the following problems (in the suggested order): (1) 3.2: 11d, 11e, 12b, 12c, 16a, 16b (2) 1.3: 4, 9, 10c, 11, 2 (3) 1.4: 1, 3, 4 Be sure to provide complete justification in grammatically correct English sentences for both Part I and Part II. Please remember to write Part II on a separate paper than Part I. Part II For the following problems, use the Principle of Mathematical Induction to prove the given statement. Carefully explain what you are doing in your proof ( e.g. your hypotheses in each step, the conclusion you wish to draw). Begin and end your proof by mentioning that you are using or have used induction. You will be graded as much on form as on mathematical content. (1) Prove that the sum of the first n odd natural numbers is n 2 . (2) Prove the power rule for derivatives of polynomials:
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Unformatted text preview: For f ( x ) = x n , f ( x ) = nx n-1 . (You may use the product rule for derivatives ( fg ) = f g + fg in your argument.) (3) Prove that the sum of the interior angles of any convex n-gon equals ( n-2)180 ◦ . (4) 1.4.7: Guess a formula for A n and prove it using induction. (5) Let A 1 ,A 2 ,...,A n be nonsingular m × m matrices. Prove that the product A 1 A 2 ··· A n is nonsingular and that ( A 1 ··· A n )-1 = A-1 n ··· A-1 1 . (*) Bonus: Prove that the Principle of Mathematical Induction and the Well-Ordering Principle (stated below) are logically equivalent. Well-Ordering Principle: Every non-empty set of positive integers has a smallest element; i.e. for any non-empty set of natural numbers S , there exists an m ∈ S such that m ≤ s for all s ∈ S . 1...
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