5. Assume x, y E R2 with ll2l1. Find a Givens rotation matrix G- CS(i.e., find c s cl (ie, find c...
(2) (a) Prove that there is a C1 map u : E → R-defined in a neighborhood E c R2 of the point (1,0) such that (b) Find u'(x) for x E E (c) Prove that there is a Cl map : G → R2 defined in a neighborhood G C R2 of the point (1,0) such that for all y EG (2) (a) Prove that there is a C1 map u : E → R-defined in a neighborhood E...
(a) Let T: R2 + R2 be counter clockwise rotation by 7/3, i.e. T(x) is the vector obtained by rotating x counter clockwise by 7/3 around 0. Without computing any matrices, what would you expect det (T) to be? (Does T make areas larger or smaller?) Now check your answer by using the fact that the matrix for counter clockwise rotation by is cos(0) - sin(0)] A A= sin(0) cos(0) (b) Same question as (a), only this time let T...
Question 2 (10 marks) Consider vectors b) (a) Show that B {bi, b2} and Ć = {ci. C2} are bases for R2 (b) Find the B-coordinates of x- (c) Find the change of coordinates matrix Pc-s from B to C and use it to find [x (d) Find the C-coordinates of y - (e) Find the change of coordinates matrix Psc from C to B and use it to find yg Question 2 (10 marks) Consider vectors b) (a) Show...
6 5.) (10 points) (a) Suppose TỈ : R2 → R2 s a reflection in the line y = z. what is A, the standard matrix of T1? (b) Suppose T : R2R is the counterclockwise rotation by 8 J. What is A2, the standard matrix of T2? (c) What is the image of (2,-4) under T- Ti o T2? 6 5.) (10 points) (a) Suppose TỈ : R2 → R2 s a reflection in the line y = z....
Find the matrix A' for T relative to the basis B'. T: R2 R2, 7(x,y) - (-9x + y, 9x - y), 8' = {(1, -1), (-1,5)} A' 11
Find the matrix A' for T relative to the basis B'. T: R2 + R2, T(x, y) = (3x - y, 4x), B' = {(-2, 1), (-1, 1)} A' = Let B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4,4)} be bases for R2, and let 0 2 A = 3 4 be the matrix for T: R2 + R2 relative to B. (a) Find the transition matrix P from B' to B. 6 4 P= 9 4...
1. Let L: R2-R2 be defined by L(x.y) (x +2y, 2x - y). Let S be the natural basis of R2 and let T = {(-1,2), (2,0)) be another basis for R2 . Find the matrix representing L with respect to a) S b) S and1T c) T and S d) T e) Find the transition matrix Ps- from T basis to S basis. f) Find the transition matrix Qre-s from S-basis to T-basis. g) Verify Q is inverse of...
(2) (a) Prove that there is a C1 map u: E → R2 defined in a neighborhood E C R2 of the point (1,0) such that (b) Find Du(x) for x є E. (c) Prove that there is a C map v G R2 defined in a neighborhood G C R2 of the point (1,0) such that for all y G. (2) (a) Prove that there is a C1 map u: E → R2 defined in a neighborhood E C...
3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin, and let S : R2 + R2 be the shear along the y-axis given by S(x,y) = (x,x+y). (You may assume that these are linear transformations.) (a) Write down, or compute, the standard matrix representations of T and S. (b) Use (a) to find the standard matrix representations of (i) SoT (T followed by S), and (ii) ToS (S followed by T). (c) Let...
For each of the following, find the standard matrix of the given transformation from R2 to R2 (a) Counterclockwise rotation through 120 about the origin. sin (a) f дх Ω (b) Projection onto the line y 5 x. sin (a) Ω да (c) Reflection in the line y= x- sin (a) Ω f