(3 pts) 7. The following matrix represents a reflection onto a line in RP: 4/5 -3/5]...
(2) The matrix A-[-5 12 12 5 represents a reflection combined with a scaling by a factor of 13. (a) Find vectors ui and v2 such that A is reflection over the line L = span(n) and such that v2 is orthogonal to vi (b) Find the eigenvectors of A with their associated eigenvalues. (c) Find an eigenbasis B for A (a basis of R2 consisting of eigenvectors of A). (d) what is the matrix of the linear transformation T(z)...
The tranformation matrix for a reflection in the line y=x is and the transformation matrixfor areflection in the line y=-x is.What is the transformation matrix for a reflection in the liney=-5/2x
5.4. Find the matrix of the orthogonal projection in R2 onto the line x1 = –2x2. Hint: What is the matrix of the projection onto the coordinate axis x1? Problem 5. Problem 5.4 on page 23. The following method is suggested: (1) Find an angle o such that the line x1 = –2x2 is obtained by rotating the x-axis by 0. (2) Convince yourself with geometry that to project a vector v onto the line x1 = –2x2 is the...
Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let P2 R2 R2 be the orthogonal projection onto the line L2: x32 2r2 0. 0. (1) What are the image and kernel of P2P What is the rank of P2P? Give a geometric description, without relying on the matrix of P2P (2) Find the matrix that represents P2P Problem 7. Let P R2 -> R2 be the orthogonal projection onto the line Li Let...
4.(5 pts)Give an example of a 3 x 3 matrix with eigenvalues of 2, 2, and -3 that is diagonalizable. Show that it is, in fact, diagonalizable, and find C and D such that C (you may make this as trivial as you wish!) AC = D 5.(5 pts) Give an example of a 3 x 3 matrix with eigenvalues of 2, 2, and -3 that is NOT diagonalizable. Show WHY it is not diagonalizable. 6. (5 pts) Let T:...
5. (8 pts) Suppose B is a 4 x 5 matrix, and the associated linear transformation T(E) Bd, is onto. (a) (3) Find dim Nul(B) (b) (2) Does Ba 5 have a unique solution for every B (c) (3) Give a geometric interpretation to the solution set of Bt- 0 5. (8 pts) Suppose B is a 4 x 5 matrix, and the associated linear transformation T(E) Bd, is onto. (a) (3) Find dim Nul(B) (b) (2) Does Ba 5...
7. (15 pts) For the matrix A= -3 1 2 3 6 -2 - 4 -9 -1 1-7 2 3 -1 5 8 - 4 4 9 0 a) Use your calculator to place the matrix in RREF. b) Find a basis for the Range(A). c) Find a basis for Nul(A).
17. The standard matrix of the the linear transformation that represents projection onto the vector 1 m onto the vector (9)}{-1 9 ®}1] (0}{-1) none of these [1 2 3] 18. The matrix O O 5 can be reduced (using elementary row operations) to [2 4 0 100] [120] 1 007 (A) 0 1 0 (B) 0 1 0 (0) 0 1 0 (D) none of these LO 0 1 LO 0 0 Lo o o 19. Which of the...
5. (3) Let X = Serie x = ()m - (*) indo 3 x 3 mais Find a 3 x 3 matrix A such that the projection PAX = e vector X ER projected onto the line I that is parallel to a and passes through the origin. Pex - Ax 6. (4) The line l in Ris given by the equation x + 3y = 0. (a) What is the angle e between the positive x-axis and the line...
Tis the reflection through the origin in RP: 7x, y) = (-X, Y), (3,2). (a) Find the standard matrix A for the linear transformation T. A- It (b) Use A to find the image of the vector v T(V) (c) Sketch the graph of vand its image. 3 T(v) 2 2 1 -3 2 - - 2 -1 2 1 3 -1 -1 -2 T(v) -31 -3 O T (v) 3 2 2 11 1