### Composition and Matrix Multiplication

Exercise Compute the composition TS

1) T(y) = 1 - y2: RR
S(x) = sin x: RR

Answer TS(x) = 1 - (sin x)2 = cos2 x: RR

2) T(x, y) = x2 + y2: R2R
S(t) = (cos t, sin t): RR2

Answer TS(t) = (cos t)2 + (sin t)2 = 1: RR

3) T(x, y) = (x + y, x - y): R2R2
S(u, v) = (1/2(u + v), 1/2(u - v)): R2R2

Answer TS(u, v) = (1/2(u + v) + 1/2(u - v), 1/2(u + v) - 1/2(u - v)) = (u, v): R2R2. The composition is identity.

4) T(x1, x2, x3, x4, x5) = (x1 - 2x2 + 3x3 - 4x4 + 5x5): R5R
S(t) = (8t, 7t, 6t, 5t, 4t): RR5

Answer TS(t) = (8t - 2×7t + 3×6t - 4×5t + 5×4t) = 12t: RR

5) T(t) = (8t, 7t, 6t, 5t, 4t): RR5
S(x1, x2, x3, x4, x5) = (x1 - 2x2 + 3x3 - 4x4 + 5x5): R5R

Answer TS(x1, x2, x3, x4, x5)
= (8(x1 - 2x2 + 3x3 - 4x4 + 5x5), 7(x1 - 2x2 + 3x3 - 4x4 + 5x5), 6(x1 - 2x2 + 3x3 - 4x4 + 5x5), 5(x1 - 2x2 + 3x3 - 4x4 + 5x5), 4(x1 - 2x2 + 3x3 - 4x4 + 5x5))
= (8x1 - 16x2 + 24x3 - 32x4 + 40x5, 7x1 - 14x2 + 21x3 - 28x4 + 35x5, 6x1 - 12x2 + 18x3 - 24x4 + 30x5, 5x1 - 10x2 + 15x3 - 20x4 + 25x5, 4x1 - 8x2 + 12x3 - 16x4 + 20x5): R5R5

6) T(x1, x2, x3) = (x1 - 2x2 + 3x3, - 2x1 + 3x2 - 4x3): R3R2
S(y1, y2) = (- 2y1 - y2, y2, y1 + y2): R2R3

Answer TS(y1, y2) = ( (- 2y1 - y2) - 2y2 + 3(y1 + y2), - 2(- 2y1 - y2) + 3y2 - 4(y1 + y2) ) = (y1, y2): R2R2. The composition is identity.

7) T(y1, y2) = (- 2y1 - y2, y2, y1 + y2): R2R3
S(x1, x2, x3) = (x1 - 2x2 + 3x3, - 2x1 + 3x2 - 4x3): R3R2