Question

LINEAR ALGEBRA: PLEASE FOLLOW THE COMMENT and please tell me what is the rotate matrix and why there is cos@ and -sin@ i think it should be cos@ and sin@ on the first row

Chapter 4.2, Problem 10E O N (2 Bookmarks) Show all steps: ON We need to find the transformation matrix L that rotates each vector by 120in the counterclockwise direction The transformation Lis given by the matrix cos θ -sin θ 0 cos θ 01 (1) Putting θ-120. in cos θ, and sin θwe first evaluate the values cos120 cos(90+30) sin 30 Again, sin 120 sin (90+30) cos 30 N/3 Hence, the transformation matrix from equation (1) will be as follows 0 0 1

For each of the following linear operators on R2,
find the matrix representation of the transformation
with respect to the homogeneous coordinate
system:

(a) The transformation L that rotates each vector
by 120◦ in the counterclockwise direction

(b) The transformation L that translates each point
3 units to the left and 5 units up

(c) The transformation L that contracts each vector
by a factor of one-third

Answer 1

The rotate matrix given by L rotates a given point (x, y) by $\theta$ in counterclockwise direction.

$L = \begin{bmatrix} cos\theta & -sin\theta &0 \\ sin\theta& cos\theta &0 \\ 0& 0 & 1 \end{bmatrix}$