Let u = (2,3), v = (-5, 6), and w = (9,0). (a) Draw these vectors...
5. (a) Let u 1,4,2), ,1,0). Find the orthogonal projection of u on v (b) Letu ,1,0), u(0,1,1), (10,1). Find scalars c,,s such that 6. (a) Find the area of the triangle with vertices , (2,0,1), (3, 1,2). Find a vector orthogonal to the plane of the triangle. (b)) Find the distance between the point (1,5) and the line 2r -5y1 (i) Find the equation of the plane containing the points (1,2, 1), (2,1, 1), (1, 1,2). 7. (a) Let...
Problem 1. The figure below shows the vectors u, v, and w, along with the images T(u) and T(v) to the right. Copy this figure, and draw onto it the image T(w) as accurately as possible. (Hint: First try writing w as a linear combination of u and v.) TV (u) Problem 2. Let u = | and v Suppose T : R2 + R2 is a linear transformation with 6 1 3) Tu = T(u) = -3 and T(v)...
Let V be the set of vectors shown below. V= [] :x>0, y>0 a. If u and v are in V, is u + v in V? Why? b. Find a specific vector u in V and a specific scalar c such that cu is not in V. O A. The vector u + v may or may not be in V depending on the values of x and y. OB. The vector u + y must be in V...
Let V be the set of vectors shown below. V= Ox>0, y>0 a. If u and v are in V, is u + v in V? Why? b. Find a specific vector u in V and a specific scalar c such that cu is not in V. a. If u and v are in Vis u + vin V? O A. The vector u + v must be in V because V is a subset of the vector space R2...
Let V be the set of vectors shown below. VE :x>0, a. If u and are in V, is u +v in V? Why? b. Find a specific vector u in V and a specific scalar c such that cu is not in V. a. If u and v are in V, is u + v in V? O A. The vector u + v may or may not be in V depending on the values of x and y....
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...
Let u and v be the vectors shown in the figure to the right, and suppose u and v are eigenvectors of a 2 x2 matrix A that correspond to eigenvalues -2 and 3, respectively. Let T: R2 R2 be the linear transformation given by T(x)-Ax for each x in R2, and let w-u+v. Plot the vectors T(u), T(v), and T(w). 2- u -2 2 4 -2 10- T(v) T(w -10 10 T(u) -10- Ay 10- T(v) T(w) T(u) 10...
41 and w be vectors, and 39-42 Properties of Vectors Let u, V, and w be ved let c be a scalar. Prove the given property. 39. u. v = v.u 40. (cu) v = c(u.v) = u • (cv) 41. (u + v). w = uw + v.w 42. (u - v)•(u + v) = | u |2 - 1 v 12
3 - 2 Let u= Note that {u, v, w} is an orthogonal set of vectors and w - -3 4 9 be a vector in subspace W, where W = Span{u, v, w}. Let y= 11 -27 Write y as a linear combination of u, v, and uw, i.e. y = ciu + cqũ + c3W. Answer: y=
(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....