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Section 5.5 Orthonormal Sets: Problem 3 Previous Problem Problem List Next Problem (1 point) -5 Use...
(1 point) -6 -3 Use Theorem 5.5.2 to write the vector v = -4 as linear...
(1 point) -6 -3 Use Theorem 5.5.2 to write the vector v = -4 as linear combination of -3/V14 1/714 -2/V13 0/V13 -3/V182 -13/V182 uj = u2 = and uz = -2/V14 3/V13 -2/V182 Note that uj, uz and uz are orthonormal. V= uj + u2+ uz Use Parseval's formula to compute ||v1|?. ||5|12=
Section 5.5 Orthonormal Sets: Problem 6 Previous Problem Problem List Next Problem 1 (1 point) Use...
Section 5.5 Orthonormal Sets: Problem 6 Previous Problem Problem List Next Problem 1 (1 point) Use the inner product < f, g >= . f(x)g(x)dx in the vector space C°[0, 1] to find the orthogonal projection of f(x) = 6x2 + 1 onto the subspace V spanned by g(x) = x - and h(x) = 1. projy(f) =
Section 5.5 Orthonormal Sets: Problem 4 Previous Problem Problem List Next Problem (1 point) Find the...
Section 5.5 Orthonormal Sets: Problem 4 Previous Problem Problem List Next Problem (1 point) Find the orthogonal projection of 11 -14 V= 9 14 onto the subspace V of R4 spanned by 5 0 2 -1 X1 = and x2 = -1 -2 4 0 projy(v) =
(1 point) Let {uj, u2, u2 ) be an orthonormal basis for an inner product space...
(1 point) Let {uj, u2, u2 ) be an orthonormal basis for an inner product space V. Suppose y = qui + buz + cuz is so that|lvl1 = V116. (v, uz) = 10, and (v. uz) = 4. Find the possible values for a, b, and c. a = CE (1 point) Suppose U1, U2, Uz is an orthogonal set of vectors in Rº. Let w be a vector in Span(v1, 02, 03) such that UjUi = 42, 02.02...
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29...
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29 ( 7 Answer the following questions and give proper explanations. (a) Is {ui, U2, uz} a basis for R3? (b) Is {ui, U2, u4} a basis for R4? (c) Is {ui, U2, U3, U4, u; } a basis for R? (d) Is {ui, U2, U3, u} a basis for Rº?! (e) Are ui, u, and O linearly independent?! Problem 6. (15 points). Let A...
(1 point) -3 10 9 Given v = 9 find the coordinates for u in the...
(1 point) -3 10 9 Given v = 9 find the coordinates for u in the subspace W spanned by 1 0 3 -3 -1 5 4 U1 = = , U2 = , U3 and 14 -7 1 Note that uj, U2, U3 and 14 are orthogonal. 2 U = U1+ U2+ U3+ 14
2 1 3 4 -2 5 7 -2 9 Problem 9 Let uj = u2 =...
2 1 3 4 -2 5 7 -2 9 Problem 9 Let uj = u2 = 13 2 Also let v= 0 5 3 10 -6 0 11 1 1 7 a) (4 pts) Compute prw(v) where W = Span{u1, U2, U3} CR5. b) [4 pts) Compute prw(v) where w+ denotes the orthogonal complement of W in R5. c) [3 pts) Compute the distance between v and W.
1- 2- 3- 1 (10 points) Show that {u1, U2, U3} is an orthogonal basis for...
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1 (10 points) Show that {u1, U2, U3} is an orthogonal basis for R3. Then express x as a linear 3 4 combination of the u's. u -3 U2 = 0 ,u3 5 6 -2 2 -1 (10 points) Suppose a vector y is orthogonal to vectors u and v. Prove that y is orthogonal to the vector 4u - 3v. 10. (2 points each) True or False: ( ) Eigenvalues must be nonzero scalars. ( )...
Problem #3: Let R4 have the inner product <u, v> = ulv1 + 2u2v2 + 3u3v3...
Problem #3: Let R4 have the inner product <u, v> = ulv1 + 2u2v2 + 3u3v3 + 40404 (a) Let w = (0,9,5,-2). Find llwll. (b) Let W be the subspace spanned by the vectors U1 = = (0,0, 2, 1), and u2 = (-3,0,–2, 1). Use the Gram-Schmidt process to transform the basis {uj, u2} into an orthonormal basis {V1, V2}. Enter the components of the vector v2 into the answer box below, separated with commas.
1Hint: Use the theorem from class that any linearly independent list of vectors is contained in...
1Hint: Use the theorem from class that any linearly independent
list of vectors is contained in a basis
2Hint: Remember that we prove the equality of sets X = Y by
showing X ⊂ Y and Y ⊂ X.
(2 points each for (a),(b),(d)) In this problem, we will prove the following di- mension formula. Theorem. If H and H' are subspaces of a finite-dimensional vector space V, then dim(H+H') = dim(H)+dim(H') - dim(H nH'). (a) Suppose {u1;...; up} is...
(1 point) -6 -3 Use Theorem 5.5.2 to write the vector v = -4 as linear combination of -3/V14 1/714 -2/V13 0/V13 -3/V182 -13/V182 uj = u2 = and uz = -2/V14 3/V13 -2/V182 Note that uj, uz and uz are orthonormal. V= uj + u2+ uz Use Parseval's formula to compute ||v1|?. ||5|12=
Section 5.5 Orthonormal Sets: Problem 6 Previous Problem Problem List Next Problem 1 (1 point) Use the inner product < f, g >= . f(x)g(x)dx in the vector space C°[0, 1] to find the orthogonal projection of f(x) = 6x2 + 1 onto the subspace V spanned by g(x) = x - and h(x) = 1. projy(f) =
Section 5.5 Orthonormal Sets: Problem 4 Previous Problem Problem List Next Problem (1 point) Find the orthogonal projection of 11 -14 V= 9 14 onto the subspace V of R4 spanned by 5 0 2 -1 X1 = and x2 = -1 -2 4 0 projy(v) =
(1 point) Let {uj, u2, u2 ) be an orthonormal basis for an inner product space V. Suppose y = qui + buz + cuz is so that|lvl1 = V116. (v, uz) = 10, and (v. uz) = 4. Find the possible values for a, b, and c. a = CE (1 point) Suppose U1, U2, Uz is an orthogonal set of vectors in Rº. Let w be a vector in Span(v1, 02, 03) such that UjUi = 42, 02.02...
1 -1.2 5 Uį = U2 = -3 1, U3 = 2 , 14 = 29 ( 7 Answer the following questions and give proper explanations. (a) Is {ui, U2, uz} a basis for R3? (b) Is {ui, U2, u4} a basis for R4? (c) Is {ui, U2, U3, U4, u; } a basis for R? (d) Is {ui, U2, U3, u} a basis for Rº?! (e) Are ui, u, and O linearly independent?! Problem 6. (15 points). Let A...
(1 point) -3 10 9 Given v = 9 find the coordinates for u in the subspace W spanned by 1 0 3 -3 -1 5 4 U1 = = , U2 = , U3 and 14 -7 1 Note that uj, U2, U3 and 14 are orthogonal. 2 U = U1+ U2+ U3+ 14
2 1 3 4 -2 5 7 -2 9 Problem 9 Let uj = u2 = 13 2 Also let v= 0 5 3 10 -6 0 11 1 1 7 a) (4 pts) Compute prw(v) where W = Span{u1, U2, U3} CR5. b) [4 pts) Compute prw(v) where w+ denotes the orthogonal complement of W in R5. c) [3 pts) Compute the distance between v and W.
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1 (10 points) Show that {u1, U2, U3} is an orthogonal basis for R3. Then express x as a linear 3 4 combination of the u's. u -3 U2 = 0 ,u3 5 6 -2 2 -1 (10 points) Suppose a vector y is orthogonal to vectors u and v. Prove that y is orthogonal to the vector 4u - 3v. 10. (2 points each) True or False: ( ) Eigenvalues must be nonzero scalars. ( )...
Problem #3: Let R4 have the inner product <u, v> = ulv1 + 2u2v2 + 3u3v3 + 40404 (a) Let w = (0,9,5,-2). Find llwll. (b) Let W be the subspace spanned by the vectors U1 = = (0,0, 2, 1), and u2 = (-3,0,–2, 1). Use the Gram-Schmidt process to transform the basis {uj, u2} into an orthonormal basis {V1, V2}. Enter the components of the vector v2 into the answer box below, separated with commas.
1Hint: Use the theorem from class that any linearly independent
list of vectors is contained in a basis
2Hint: Remember that we prove the equality of sets X = Y by
showing X ⊂ Y and Y ⊂ X.
(2 points each for (a),(b),(d)) In this problem, we will prove the following di- mension formula. Theorem. If H and H' are subspaces of a finite-dimensional vector space V, then dim(H+H') = dim(H)+dim(H') - dim(H nH'). (a) Suppose {u1;...; up} is...
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