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8) Is B = {(1,-1,2), (0,2,1),(5,1,-2)} an orthonormal basis of R3? If it's not, can we...
1.1. Sample Problems. 1. In R3, (a) Find an orthonormal basis which contains the vector ( 1, 0); (b) Find the component of the vector (1,1,1) in the direction of (1,0,1).
17 Find the orthogonal complement of the following. a. U = sp({(3,-1,2)}) in R3. b. V=({(1,3,0), (0,2,1))) in R3. Do this both algebraically and geometrically. Compare with part a. c. W=sp({1+x}) in 81 (-1,1]).
(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...
5. Consider the orthonormal basis B = {b, bs.bu) = {* B= {b1,b2, -11 !} for R3 Orthonormal just means b; · b; is 0 unless i = j, in which case it is 1. [ 21 (a) Let v= | -1 . Caculate the dot products: a=v.bi, b=v.b2, c=v.b3. (1) Show that lolo = [:] (c) Will this always work?
Find an ONB (orthonormal basis) for the following plane in R3 2 + y + 3z = 0 First, solve the system, then assign parameters s and t to the free variables (in this order), and write the solution in vector form as su + tv. Now normalize u to have norm 1 and call it ū. Then find the component of v orthogonal to the line spanned by u and normalize it, call it ū. Below, enter the components...
(11 Let u Show that B } is an orthogonal basis of R3. (b) Convert B into an orthonormal basis C of R3 by normalizing ü, ū and w. Show your work. Find the change of coordinates matrices Psee and Pee-swhere C is the or- thonormal basis of R3 you found in (b) and S is the standard basis of R3. Justify your answers. Suppose now that ü, ū and w are eigenvectors of a 3 x 3 matrix A...
3 The two vectors X1 = 0 -1 8 X2 = 5 -6 form a basis for a subspace w of Rº. Use the Gram-Schmidt process to produce an orthogonal basis for W, then normalize that basis to produce an orthonormal basis for W.
4. Let R3 be equipped with the dot product and let B = {(2,1,1),(0,4,-2),(3,5, -1)}. Bis a basis for R3. Use the Gram-Schmidt process to convert B into an orthonormal basis.
Please attempt both questions. 5. Find an orthonormal basis for the plane viewed as a subspace of R3. Z (-1,0,2) (0,-1,0) (0,1,0) X 6. Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): 1 2 5 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = 22 - 3, 9() = 4, h(x) = 2² +2}...
4. Let R3 be equipped with the dot product and let B = {(2,1,1),(0,4,-2),(3,5, -1)}. B is a basis for R?. Use the Gram-Schmidt process to convert B into an orthonormal basis.