3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin,...
(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 19: Linear Transformations Let S = {(u, v): 0 <u<1,0 <v<1} be the unit square and let RCR be the parallelogram with vertices (0,0), (2, 2), (3,-1), (5,1). a. Find a linear transformation T:R2 + R2 such that T(S) = R and T(1,0) = (2, 2). What is T(0, 1)? T(0,1): 2= y= b. Use the change of variables theorem to fill in the appropriate information: 1(4,)dA= S. ° Sºf(T(u, v)|Jac(T)| dudv JA JO A= c. If f(x, y)...
(b) Let F: R2 + Rº be a vector field on R2 defined as F(x, y) = (3y, 22 – y). Suppose further that ^ C R2 is a curve in R2 consisting of the parabola y = 22 - 1 for 1 € (-1,0) and the straight line y = 1 – 1 for 1 € [0,1]. (i) Sketch the curvey in R2 [2] (ii) By considering the curve y piecewise, compute the vector field integral: [5] F(x). F(x)...
(c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S onto S' (0,1) v-axis V=1 (2,1) (1,1) y =(x-1)2 у-ахis u 1 v=u-1 u-axis (1,0) (0,0) х-аxis (1,0) (c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S...
(3) Let ф : R2-> R2 be 90° counter-clockwise rotatation about the origin. (a) Find the matrix which A represents ф with respect to the standard basis. (b) what the the eigenvalues and eigenvectors of67 (c) If we consider A to be a complex matrix (since all real numbers are complex numbers), what are the eigenvalues and eigenvectors of A? (3) Let ф : R2-> R2 be 90° counter-clockwise rotatation about the origin. (a) Find the matrix which A represents...
Q3 (6 points) Let T be the rotation (counterclockwise) of R2 by an angle ofLet S R2R2 be the reflection with respect to the line y 3 x. S and T are both linear transformations. (a) (4 points) Determine the standard matrix of S. Give details about how your obtained your answer. b) (1 point) Write down the standard matrix of T. Give only the answer, no justification needed. (c) (1 point) What is the standard matrix of S。TY Drag...
Let T:R? → Rº be the linear transformation that first rotates points clockwise through 30° (7/6 radians) and then reflects points through the line y = 2. Find the standard matrix A for T. A=
(1 point) Let S = {1, 2, 3} and T : Fun(S) + Rº be the transformation T(f) = (f(2) – 2 f(1), f(2) + f(3), f(1)) and consider the ordered bases E = {x1 X1, X2, X3 > the standard basis of Fun(S) F = {xı – X3, 2X1 + X2, X3 – x2} a basis of source Fun(S) E' = {(1,0,0), (0,1,0), (0,0,1)}the standard basis of R3 G = {(-2, –1,1), (1,-1,0), (0,1,0)} a basis of target R3...
Let T : R2 → R2 be the linear transformation given by T(v) = Av that consists of a counterclockwise rotation about the origin through an angle of 30 2, Find the matrix that produces a counterclockwise rotation about the origin through an angle of 30°. Be sure to give the EXACT value of each entry in A. a. b. Plot the parallelogram whose vertices are given by the points A(0, 0), B(4, 0), C(5, 3), and D 1, 3)...
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove İ}!").lx (ly İ)(2 dr dy. Pdr+Qdy That is, prove Green's Theorem for the triangle A. [Hint: the lecture notes have a proof for when A is a rectangle. So, the idea is is to give a similar proof where we have this...