(4) Recall that Rot(P,) denotes rotation about a point P through angle e counterclockwise. Let ABCD...
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)...
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....
Thanks ! A rotation through an angle of about the point (5,4). Find the image of the triangle having the following vertices A(1,2), B(2,8), and C(3,2) under this rotation.
(c) [1 point] Let R : E3 → E3 be the rotation in E3 with axis in the direction of the vector ã=(-1,2, -2) and angle 0 = . If pe E3 denotes the point (0,0,1) then ... R(p) = (d) [1 point] Let R: E2 → Eº be a reflection through a line l that fixes the origin and sends (1,1) to some point on the line y = x. Can you determine the line l? If so, give...
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...
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
(1 point) Match each linear transformation with its matrix. A. Contraction by a factor of2 B. Rotation through an angle of 90 in the clockwise direction C. Projection onto the y-axis D. Reflection in the y-axis E. Rotation through an angle of 90° in the counterclockwise direction -1 0 0.5 0 0 0.5 0 -1 F. Reflection in the r-axis 0 -1 (1 point) Match each linear transformation with its matrix. A. Contraction by a factor of2 B. Rotation through...
(4) (a) Determine the standard matrix A for the rotation r of R 3 around the z-axis through the angle π/3 counterclockwise. Hint: Use the matrix for the rotation around the origin in R 2 for the xy-plane. (b) Consider the rotation s of R 3 around the line spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a basis of R 3 for which the matrix [s]B,B is equal to A from (a). (c) Give...
If U(,) refers to a rotation through an angle ß about the y-axis, show that the matrix elements (j, m|U(B, Ý)|j, m'), -ism, m' Sj, are polynomials of degree 2j with respect to the variables sin (6/2) and cos (B/2). Here [j, m) refers to an eigenstate of the square and z-component of the angular momentum: j2|j, m) = jlj +1) ħaj, m), İzlj, m) = mħ|j, m).
In this problem, let li be the line that passes through the points A(1,2, 4) and B(-1,3,8), and let l2 be the line with symmetric equations x +1 = 2y = 32 — 3. Parts (e) and (f) relate to the vector field F = (xy, xz, yz). (a) Show that the lines li and l2 intersect. (b) Let P be the plane that contains both lines li and lz. Find an equation for P. (c) Show that the points...