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1 (10 points) Show that {u1, U2, U3} is an orthogonal basis for...
10. (2 points each) True or False: ( ) Eigenvalues must be nonzero scalars. The sum...
10. (2 points each) True or False: ( ) Eigenvalues must be nonzero scalars. The sum of two eigenvectors of a matrix A is also an eigenvector of A. ( ) If A is diagonalizable, then the columns of A are linearly independent. () If r is any scalar, then ||rv|| = r|| ||. () The length of every vector is a positive number.
7. (10pts) Show that {u1, U2, u3}is an orthogonal basis for R°. Then express x as...
7. (10pts) Show that {u1, U2, u3}is an orthogonal basis for R°. Then express x as a linear combination of the u's. -i]--}-|--}) [:] and x = , U3 = , U2 = 4 1
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. If U1, U2, U3 are i.i.d. Unif(0,1), what’s the distribution of ? 2. If U...
1. If U1, U2, U3 are i.i.d. Unif(0,1), what’s the distribution of ? 2. If U and V are i.i.d. Unif(0,1), what’s the distribution of + ? -3ln(U1(1- U2)(1 - U3)) -2 cos(2TV)-In(U)) n(U) sin(2T V) -3ln(U1(1- U2)(1 - U3)) -2 cos(2TV)-In(U)) n(U) sin(2T V)
(1 point) 0 Given v 3 find the linear combination for v in the subspace W spanned by 0 0 3 3 and 114 , u2 = , из- 4 4 Note that ul , u2 , u3 and 14 are orthogonal. u1+ 7 U2 ll4 (1 point) 0 Given...
(1 point) 0 Given v 3 find the linear combination for v in the subspace W spanned by 0 0 3 3 and 114 , u2 = , из- 4 4 Note that ul , u2 , u3 and 14 are orthogonal. u1+ 7 U2 ll4
(1 point) 0 Given v 3 find the linear combination for v in the subspace W spanned by 0 0 3 3 and 114 , u2 = , из- 4 4 Note that ul...
Show that (u, u2 ub) is an orthogonal basis for R Thon express as a inear...
Show that (u, u2 ub) is an orthogonal basis for R Thon express as a inear of the u's Which of the following cniteria are necessary for a set of vectors to be an orthogonal basis for a subspace W of IR? Select all that apply A. The vectors must span W B. The vectors must all have a length of 1 D C. The distance between any pair of distinct vectors must be constant D. The vectors must form...
[10 pointsjConsider an orthogonal matrix Q, which has two nonzero orthogonal eigenvectors v1 and ...
[10 pointsjConsider an orthogonal matrix Q, which has two nonzero orthogonal eigenvectors v1 and v2 whose corresponding eigenvalues are λι = 3 and λ2-4, respectively. Now consider a vector y = Vi + vȚvayı + λ2V2 and compute 1QTQQy in terms of the eigenvectors and eigenvalues of Q 4.
[10 pointsjConsider an orthogonal matrix Q, which has two nonzero orthogonal eigenvectors v1 and v2 whose corresponding eigenvalues are λι = 3 and λ2-4, respectively. Now consider a vector y =...
(10 points) Verify that {u, uz} is an orthogonal set, and then find the orthogonal projection...
(10 points) Verify that {u, uz} is an orthogonal set, and then find the orthogonal projection -4 of y onto Span{u1, u2}. y --8) 3
#12 6.3.20 s Question Help 5 0 Let un 2. u2 -8 and uz = 1...
#12
6.3.20 s Question Help 5 0 Let un 2. u2 -8 and uz = 1 Note that u, and uz are orthogonal. It can be shown that ug is not in the subspace W spanned by u, and up. Use this to - 1 0 construct a nonzero vector v in R3 that is orthogonal to u, and up. 4 The nonzero vector v = is orthogonal to u, and u2
(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
10. (2 points each) True or False: ( ) Eigenvalues must be nonzero scalars. The sum of two eigenvectors of a matrix A is also an eigenvector of A. ( ) If A is diagonalizable, then the columns of A are linearly independent. () If r is any scalar, then ||rv|| = r|| ||. () The length of every vector is a positive number.
7. (10pts) Show that {u1, U2, u3}is an orthogonal basis for R°. Then express x as a linear combination of the u's. -i]--}-|--}) [:] and x = , U3 = , U2 = 4 1
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) 0 Given v 3 find the linear combination for v in the subspace W spanned by 0 0 3 3 and 114 , u2 = , из- 4 4 Note that ul , u2 , u3 and 14 are orthogonal. u1+ 7 U2 ll4
(1 point) 0 Given v 3 find the linear combination for v in the subspace W spanned by 0 0 3 3 and 114 , u2 = , из- 4 4 Note that ul...
Show that (u, u2 ub) is an orthogonal basis for R Thon express as a inear of the u's Which of the following cniteria are necessary for a set of vectors to be an orthogonal basis for a subspace W of IR? Select all that apply A. The vectors must span W B. The vectors must all have a length of 1 D C. The distance between any pair of distinct vectors must be constant D. The vectors must form...
[10 pointsjConsider an orthogonal matrix Q, which has two nonzero orthogonal eigenvectors v1 and v2 whose corresponding eigenvalues are λι = 3 and λ2-4, respectively. Now consider a vector y = Vi + vȚvayı + λ2V2 and compute 1QTQQy in terms of the eigenvectors and eigenvalues of Q 4.
[10 pointsjConsider an orthogonal matrix Q, which has two nonzero orthogonal eigenvectors v1 and v2 whose corresponding eigenvalues are λι = 3 and λ2-4, respectively. Now consider a vector y =...
(10 points) Verify that {u, uz} is an orthogonal set, and then find the orthogonal projection -4 of y onto Span{u1, u2}. y --8) 3
#12
6.3.20 s Question Help 5 0 Let un 2. u2 -8 and uz = 1 Note that u, and uz are orthogonal. It can be shown that ug is not in the subspace W spanned by u, and up. Use this to - 1 0 construct a nonzero vector v in R3 that is orthogonal to u, and up. 4 The nonzero vector v = is orthogonal to u, and u2
(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
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