Latin = and l [1 + 3i -2i (a) Verify that ởi and ū2 are orthogonal....
1. Show that {ū1, ū2, ū3} is an orthogonal basis for R3, and write ž as a linear combination of the vectors {ū1, ū2, ū3}, 1 ū1 -[:] -2 -2 ū2 = [ ] ū3 = Ž -11 -3 -17
1. (50 pts.) Let A be the 3 x 3 matrix A= 0 0 3 0 2 0 3 0 0 :) i. Compute the eigenvectors ū1, U2, U3 of A. ii. Verify that the matrix S with columns ū ū2, öz has full rank. iii. Use the Gram-Schmidt process to change B into an orthogonal matrix P.
3. Consider two vectors u = 2i -j +2k and v=3i+2j-k. (a) Find a vector orthogonal to a and b. _ [3 marks] (b) Show that the vector from (a) is orthogonal to a and b. [1 mark]
Let W = Span{ū1, ū2}. Write y as the sum of a vector We W and a vector zew, 1 0 -2 17 -11 3 ū1 = 2 y= 2 0 2
Verify that (u,,uz) is an orthogonal set, and then find the orthogonal projection of y onto Span (u.uz). 1-17 [3] 2,,= -1 . uz = = To verify that (uy,uz) is an orthogonal set, find u. U. Uyuz = 0 (Simplify your answer.) The projection of y onto Span{u,, 42} is (Simplify your answers.)
Verify that {u7,42} is an orthogonal set, and then find the orthogonal projection of y onto Span{uq, 42}- 6 3 - 4 y- . 01 u:- -2 0 To verify that (14,42} is an orthogonal set, find uy • 42. u uy - (Simplify your answer.) The projection of y onto Span{44,42} is .. (Simplify your answers.)
Verify that (41.uz) is an orthogonal set, and then find the orthogonal projection of y onto Span{41.42}- y = 1 0 To verify that {0, 42} is an orthogonal set, find u, '42. u U2 - 0 (Simplify your answer) The projection of y onto Span{u, uz} is (Simplify your answers)
2. Which of the following pairs of vectors are orthogonal? (a) v = 3i - 2j, w = --i +2j (b) v = -2i, w = 5j (c) v = -i + 2j, w = -1 (d) v = 2i – 3j, w = -2i + 3j (e) None of these
(i) Find an orthonormal basis {~u1, ~u2} for S (ii) Consider the function f : R3 -> R3 that to each vector ~v assigns the vector of S given by f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a linear function. (iii) What is the matrix of f in the standard basis of R3? (iv) What are the null space and the column space of the matrix that you computed in the previous point? Exercise 1. In...
(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