HAPPY STUDYING
17 Find the orthogonal complement of the following. a. U = sp({(3,-1,2)}) in R3. b. V=({(1,3,0),...
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...
2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector v in the orthogonal complement of the space 0 Span 2,2 Do not simply compute the cross product. (c) Let A be a 5 × 2 rnatrix with linearly independent columns. Using the rank-nullity theorem applied to AT, and any other results from the course, find the dinension of Col(A) 2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector...
(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....
(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...
Caculate w=2U+V, |U|,|V| and |W| A= <1,-3,7> and B=<-5,2,-3> Find C sp A+B+C=0
3. (6 marks) Suppose that u, v and w are vectors in R3, and that u. (v x w) = 3. Determine (a) u (w x v) (b) u: (w X w) (c) (2u x v). w
Consider the points u(1, 1,-1), v (a, 2,-1) and w (1,2,-1) in R3, where a e R. There are two possible values of a for which u, v and w will form an isosceles triangle. a) Find one of these values. (b) Determine the angle between the equal sides of the triangle. Consider the points u(1, 1,-1), v (a, 2,-1) and w (1,2,-1) in R3, where a e R. There are two possible values of a for which u, v...
0 17 (2 points) Find the projection of5onto the subspace W of R3 spanned by6 U- -1 projw (V) 0 17 (2 points) Find the projection of5onto the subspace W of R3 spanned by6 U- -1 projw (V)
4.4.3. Find the orthogonal projection of v (1,2,-1,2) onto the following subspaces: 12 20 1-1 01 (a) the span of2 (b) the ma of the aris b3(0) the kernel of the matrix-2 Warning. Make sure you have an orthogonal basis before applying formula (4.42)! ; (d) the subspace orthogonal to a (1,-1,0,1) 4.4.3. Find the orthogonal projection of v (1,2,-1,2) onto the following subspaces: 12 20 1-1 01 (a) the span of2 (b) the ma of the aris b3(0) the...
(6 marks) Suppose that u, v and w are vectors in R3, and that u. (v x w) = 3. Determine (a) u (w xv) (b) u. (w xw) (c) (2u x v). w