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 countercloc...
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....
Assume that T is a linear transformation. Find the standard matrix of T. T: R2→R2, rotates points (about the origin) through-6 radians. Type an integer or simplified fraction for each matrix element. Type exact answers, using radicals as needed.)
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...
Assume that T is a linear transformation. Find the standard matrix of T... Assume that T is a linear transformation. Find the standard matrix of T 2T radians T: R2 R2, rotates points (about the origin) through 3 A = (Type an integer or simplified fraction for each matrix element. Type exact answers, using radicals as needed.)
In R2 let R be the rotation about the origin through the angle 27/14. Then the matrix [R] representing Ris [R] = Consequently R transforms the point (1, 2) into Check
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
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...
For each of the following, find the standard matrix of the given transformation from R2 to R2. (a) Clockwise rotation through 30° about the origin. a ab sin(a) 22 ar (b) Projection onto the line y = -42. a ab sin(a) !!! 22 8 (c) Reflection in the line y = 1 a ab sin(a) 22 ? Әr
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...
(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...