Let TA R2 right R2 be a linear transformation that rotates p

Let T_A: R^2 right R^2 be a linear transformation that rotates points about the origin through pi/3 radians (counterclockwise) and let T_B: R^2 be a linear transformation that reflects points through the line x_2, = -x_1, and let X_0 = [2 2]. a. Find the standard matrix A such that T_A(x) = Ax and find the standard matrix B such that T_B(x) = Bx. b. Let y = T_A(X_0) and let z = T_B(y). What are y and z? c. Now let y = T_B(x_0) and let z = T_A (y). What are y and z now? Is the z here the same as the z in part (b)? d. Let T_AB(x) = ABx and let T_BA(x) = BAx. What are T_AB (x_0) and T_BA (x_0)? Compare these to the z\'s you calculated in (b) and (c) above. What property of matrix arithmetic does this demonstrate? In a small town the residents have only three choices when it comes to television viewing; they can subscribe to cable, they can pay for satellite service, or they watch no TV. (The town Is sufficiently remote so that an antenna does not

Solution

3. (a) . The matrix representing counterclockwise rotation about the origin by an angle is A =

cos

-sin

sin

cos

Here = /3, so that A =

cos /3

-sin /3

sin /3

cos /3

or, A =

1/2

- 3/2

3/2

1/2

The matrix representing reflection across the line x₂ = mx₁ is B =

1-m²

2m

2m

m²-1

Here, m = -1 so that B =

0

-2

-2

0

(b) If y = T_A(x₀), then y = Ax₀= (1-3,1+3)^T. Also, z = T_B(y) = By = (-2-23,-2+23)^T.

(c ) If y = T_B(x₀), then y = Bx₀ = (-4,-4)^T. Also, z= T_A(y) = Ay= (-2+23,-2-23)^T. No, z, here, is not same as the z in part(b).

(d) T_AB(x₀) = AB.x₀= (-2+23, -2-23)^T and T_BA(x₀) = BA.x₀= (-2-23,-2+23)^T. On comparing these with z in part (b) and (c), we observe that T_B(T_A(x₀)) = T_BA(x₀) and T_A(T_B(x₀)) = T_AB(x₀). The property of the matrix arithmetic conveyed here is ABBA.

cos	-sin
sin	cos