1. Find the coordinates/coefficients of 4 in terms of the basis G) 3 and u2 =
. (4 points) Use the Gram-Schmidt process to transform the basis fui u2) wher«e u (1,-3), u2 (2,2) into an orthormal basis for R2. Draw both sets of basis vectors in the ry-plane
(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
(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...
Prove the following: (a) Let V be a vector space of dimension 3 and let {v,U2,U3} be a basis for V. Show that u2, u2 -2+s and uvi also form a basis for V (b) Show that1-,1-2,1-- 2 is a basis for P2[r], the set of all degree 2 or less polynomial functions. (c) Show that if A is invertible, then det A (Note: Show it for any det A-1 square matrix, showing it for a 2 x 2 matrix...
(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...
Suppose that the columns of A form a basis of R4. Find the coordinates of x relative to basis A. (Note that only the inverse of A has been given) 1 0 2 A-1 2 -3 5 1 2 5 4 6 3 0 -21 X = -1 1 1 0
10. Let S be a regular surface with E = G = (1 + u2 + U2)2, F = 0 and e = 2=-g,f=0. (a) Find the Gaussian and mean curvatures b)Find the principal curvatures and directions of S
10. Let S be a regular surface with E = G = (1 + u2 + U2)2, F = 0 and e = 2=-g,f=0. (a) Find the Gaussian and mean curvatures b)Find the principal curvatures and directions of S
Suppose that the columns of A form a basis of R4. Find the coordinates of x relative to basis A. (Note that only the inverse of A has been given) 2 1 2 5 -3 4 0 7 2 A=1 - x= 5 -1 6 0 1 1 1 -21 0
Exercise 1: (20pts) Let u1-11, 1, 1)T, u2-(1, 2, 2)T, u,-(2, 3, 4)T, ν,-(4,6,7)T, v2 = (0, 1,1)1 , V3 = ( ) (a) Find the transition matrix from fvi, v2, vs] to sui, u2, us] (b) If x 2vı +3v2 - 4vs, determine the coordinates of x with respect to fui, u2, us] 0,1,2
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...