Problem #5: (a) Let u =(2, -4,-8, -10) and v=(-1, -3, 8, -10). Find ||u –...
Problem #5: (a) Let u = (10, 4, -1, 8) and v = (-5, 10, -6, -1). Find ||u - proj,u||. Note: You can partially check your work by first calculating proj^u, and then verifying that the vectors projyu and u proj^u are orthogonal (b) Consider the following vectors u, v, w, and z (which you can copy and paste directly into Matlab). u -8.9 -9.8 -5.8], v = [0.8-4.1 -3.71, w = [8.6 -9.1 -8.11, [3.4 4.8 3.11 z...
Let u = [1, 3, -2], v = [-1, 1, 1], w = [5, 1, 4]. a) Check if the system of vectors {u,v,w} is an orthogonal or othonormal basis of E3. b) Find the coordinates of the vector [1,0,1] in this basis.
Find the best approximation to z by vectors of the form C7 V + c2V2. 3 1 3 -1 -6 1 z = V2 4 0 -3 3 1 The best approximation to z is . (Simplify your answer.) - 15 - 8 8 - 1 Let y = , and v2 Find the distance from y to the subspace W of R* spanned by V, and vą, given 1 0 1 - 15 3 3 - 13 09 that...
Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.
Exercise Set Chapter 3 Q1) Let u = (2, -2, 3), v = (1, -3, 4), and w=(3,6,-4). a) Evaluate the given expression u + v V - 3u ||u – v| u. V lju – v|w V X W ux (v x W) b) Find the angle 8 between the vector u = (2,-2,3) and v = (1, -3,4). c) Calculate the area of the parallelogram determined by the vector u and v d) Calculate the scalar triple product...
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...
#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) Let u = (-1,2) and v = (3, 1). (a) (5] Find graphically the vector w = (2u - v). (b) (5] Find algebraically the vector z=3u - 2 (2) (a) [5] Write u ='(1, -5, -1) as a linear combination of v1 = (1,2,0), v2 = (0,1,-1), V3 = (2,1,1). (b) (5] Are the 4 vectors u, V1, V2, V3 linearly independent? Explain your answer. (C) (5) Are the 2 vectors V, V3 linearly independent? Explain your answer....
Problem #7: Which of the following statements are always true for vectors in R3? (i) If u (vx w)-4 then w - (vxu)-4 (ii) (5u + v) x (1-40 =-21 (u x v) (ili) If u is orthogonal to v and w then u is also orthogonal to w | V + V W (A)( only (B) (iii) only (C) none of them (D) (i) and (iii) only (E) all of them (F) (i) only (G)i and (ii) only (H)...
2) Let 4 =(0,5), 4, =(-3, -1) v; = (1,-5), v, =(-2,2) and let L be a linear operator on R? whose matrix representation with respect to the ordered basis {u, uz} is 2 A= a) Determine the transition matrix S) (change of basis matrix) from {v, v,} to {u,,u,} (Draw the commutative triangle). I