Homework 7 Solutions

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$Math 33B/1 S00 K. Solutions to Midterm Exam G1, Version B If you ...$

$Math 3C Homework 3 Solutions$

Solution: Since β is a spanning set, we may write x = n i=1 cizi, with all zi ∈ β. Then 〈x, x〉 = x, n i=1 cizi = Coercivity of the inner product implies that x = 0. b. Prove that if 〈x, z〉 = 〈y, z〉 for all z ∈ β, then x = y. n ci 〈x, z〉 = 0. Solution: If 〈x, z〉 = 〈y, z〉, then 〈x − y, z〉 = 〈x, z〉 − 〈y, z〉 = 0 for all z ∈ β. By the above, x − y = 0, so x = y. 10. Let V be an inner product space, and suppose that x and y are orthogonal vectors in V. Prove that x + y 2 = x 2 + y 2 . Deduce the Pythagorean theorem in R 2 . Solution: By orthogonality of x and y, i=1 x + y 2 = 〈x + y, x + y〉 = 〈x, x〉 + 〈x, y〉 + 〈y, x〉 + 〈x, y〉 . = 〈x, x〉 + 〈y, y〉 = x 2 + y 2 . In R 2 , if two orthogonal vectors x and y are laid head-to-tail, they form two edges of a right triangle, of which x + y forms the third edge. If we denote the lengths of these edges by a, b, and c, respectively, then Thus we prove the Pythagorean Theorem. c 2 = x + y 2 = x 2 + y 2 = a 2 + b 2 . 12. Let {v1, v2, . . . , vk} be an orthogonal set in V, and let a1, a2, . . . , ak be scalars. Prove that k i=1 Solution: Clearly, in the case that k = 1, k i=1 aivi 2 aivi 2 = k i=1 |ai| 2 vi 2 . = a1v1 2 = |a1| 2 v1 2 = k i=1 |ai| 2 vi 2 . Now, assume the result is proven for sets of k −1 or fewer orthogonal vectors. Suppose v1, . . . , vk are orthogonal, and define bk−1 = k−1 i=1 aiki. Observe that k−1 k−1 〈bk−1, ak〉 = = ai 〈vi, vk〉 = 0, i=1 aivi, akvk 6 i=1
so ak and bk−1 are orthogonal. Applying the previous problem, By assumption, k i=1 aivi 2 = akvk + bk−1 2 = |akvk 2 + bk 2 . = |ak| 2 vk 2 k−1 + |ai| 2 vi 2 = i=1 k i=1 |ai| 2 vi 2 . 17. Let T be a linear operator on an inner product space V, and suppose that T(x) = x for all x. Prove that T is one-to-one. Solution: If v ∈ N(T), then v = T(v) = 0 = 0. By coercivity of the norm, v = 0. Thus T is one-to-one. 7
Page 1 and 2: 5.2 - Diagonalizability Homework 7
Page 3 and 4: Thus β = {(1, 0, 1), (0, 1, 0), (1
Page 5: Solution: Let β be a basis of V th

Homework 7 Solutions

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