Q3 (6 points) Let T be the rotation (counterclockwise) of R2 by an angle ofLet S...
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)...
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....
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p : R2 → R2 be defined similarly for the angle φ. (a) (8 points) Find the standard matrices for the linear transformations To and To. That is, let A be the matrix associated with Tip, and let B be the matrix associated with To....
Q3 (3 points) Show that if both AB and B A are defined then AB and BA are square matrices. + Drag and drop your images or click to browse... Q4 (3 points) Let A = (a) be a 2 x 2 matrix. The trace of A. which we denote by tr(A) is a number defined as tr(A) = 0 + 0x2. Prove the following properties of this number for 2 x 2 matrices A and B and a real...
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...
Question 21 Let T:R+ R2 be the transformation of rotation counterclockwise by 30 degrees, and let A be the standard matrix for T. Which one of the following statements must be true about A? A has one real eigenvalue with algebraic multiplicity 2. A has two distinct real eigenvalues. A has one real eigenvalue and one complex eigenvalue. A has two distinct complex eigenvalues. Question 22 Suppose A is a positive stochastic 2 x 2 matrix and A As n...
Problem 3. Let T R2 -R be a linear transformation, with associated standard matrir A. That is [T(TleAl, where E = (e1, ē2) is the standard basis of R2. Suppose B is any basis for R2 a matrix B such that [T()= B{v]B. This matric is called the the B-matrix of T and is denoted by TB, (2) What is the first column of T]s (3) Determine whether the following statements are true or (a) There erists a basis B...
: Let L: P1 → Pz be defined by L[p(t)] = t_p(t). Let S {t, 1} and S' = {t,t +1} be bases for P1. Let T = {t”, t², t, 1} and T' {tº, t? – 1,7,t+1} be bases for P3. Find the matrix of L with respect to (a) S and T and (b) S' and T + Drag and drop your files or click to browse...
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Q2 (6 points) If C is a smooth plane curve with parametrization r r(t),t E [a, b], then the curvature K(t) of C at the point r(t) is defined to be the magnitude of the rate of change -ll dT of the unit tangent vector with respect to the arc length. That is, = ds () [2p] Show that K(t) = ||F (C) xr" (t)|| r...
(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...