Let w-span( (1,2,0), (2,-1,1)). (i) Project y = (1,0,3) orthogonally onto W. (ii) what is the...
Let u = (2,-1,1), v= (0,1,1) and w = (2,1,3). Show that span{u+w, V – w} span{u, v, w} and determine whether or not these spans are actually equal.
7. Let W = Span{x1, x2}, where x1 = [1 2 4]" and X2 – [5 5 5]" a. (4 pts) Construct an orthogonal basis {V1, V2} for W. b. (4 pts) Compute the orthogonal projection of y = [0 1]' onto W. C. (2 pts) Write a vector V3 such that {V1, V2, V3} is an orthogonal basis for R", where vi and v2 are the vectors computed in (a).
3 y+ z 0 2. Let W be a plane characterized by the equation W. D (5 Find an orthonormal basis for (57) Find the standard matrix for the orthogonal projection of R onto W 2) Find the distance between a vector (2, 2, 15) and the plane W. (5 (3
3 y+ z 0 2. Let W be a plane characterized by the equation W. D (5 Find an orthonormal basis for (57) Find the standard matrix for the...
Let W = Span{ū1, ū2}. Write y as the sum of a vector We W and a vector zew, 1 0 -2 17 -11 3 ū1 = 2 y= 2 0 2
5 3 1 0 Problem 10 Let wi = ,W2 W3 Let W = Span{W1,W2, W3} C R6. 11 9 1 2 a) [6 pts] Use the Gram-Schmit algorithm to find an orthogonal basis for W. You should explicitly show each step of your calculation. 10 -7 11 b) [5 pts) Let v = Compute the projection prw(v) of v onto the subspace W using the 5 orthogonal basis in a). c) (4 pts] Use the computation in b) to...
- 21 For the set of vectors B= let H= span B. 2 13 For y = find the vector in H that is closest to y (note y&H) u N The vector in H that is closest to y is y=[
2. (a) Let T be the linear transformation which projects R3 orthogonally onto the plane 2x+3y+4a-0. what are the eigenvalues and associated eigenspaces of T? Justify your answer (b) Does the linear transformation described in (a) have an inverse? Why, or why not? [10 pts]
2. (a) Let T be the linear transformation which projects R3 orthogonally onto the plane 2x+3y+4a-0. what are the eigenvalues and associated eigenspaces of T? Justify your answer (b) Does the linear transformation described in...
2 2 2 Let y = 6,41 . - uz = کہانی and W = Span {uq,42}. Complete parts (a) and (b). 1 WN w UTUS a. Let U = = [u un uz]. compute UTU and UU! and UUTA (Simplify your answers.) b. Compute projwy and (UT)y. projwy = and (uu)y=(Simplify your answers.)
Let S (2,0, 1), 2- (1,2,0),s (1, 1, 1)) and J- (w (6,3,3), w (4,-1,3),u3 (5,5, 2)] be two bases of R3 Forv E R3 let (z, z2,73) and (1s) be the coordinates of v with respect to the bases T and S, respectively. u72 a) Compute the matrix giving the change of coordinates from the J-basis to the S-basis, i.e., determine the matrix A so that - Ay if x and y are as above. b) Ify (1, 0,...
Q2. x+y (a). Let f(x,y) = x²+y²+1 Find (i). lim (x,y)-(1,1) f(x,y) (ii). lim f(x,y) (x,y)-(-1,1) (iii). lim f(x,y) (x,y)-(1,-1) (iv). lim f(x,y) (x,y)-(0,0) ( 4x²y if (x, y) = (0,0) Q3. Let f(x,y) = x2 + y2 1 if (x,y) = (0,0) Find (i). lim f(x,y) (x,y)--(0,0) (ii). Is f(x,y) continuous at (0,0)? (iii). Find the largest set S on which f(x,y) is continuous.