a The linear transformation T1 R2 rightarrow R2 is given by

a. The linear transformation T_1: R^2 rightarrow R^2 is given by: T_1 (x, y) = (2x + 11 y, 10x + 56y). Find T^-1_1 (x, y) T6-1_1(x, y) = (28 x + -1y, -5x + 1y) b. The linear transformation T_2: R^3 rightarrow R^3 is given by: T_2(x, y, z) = (x + 2z, 2x + y, 2y + z). Find T^-1_1 (a, y, z). c. Using T_1 from part a it is given that: T_1(x, y) = (5, -4) Find x and y. d. Using T_2 from part b, it is given that: T_2 (x, y, z) = (6, -5, -1) Find x, y, and z.

Solution

(a) The matrix for the linear transformation T₁ is A =

2

11

10

56

Then A^-1=

28

-11/2

-5

1

Hence T₁^-1(x,y) = (28x -11y/2, -5x+y)

(b) The matrix for the linear transformation T₂ is A =

1

0

2

2

1

0

0

2

1

Then A^-1=

1/9

4/9

-2/9

-2/9

1/9

4/9

4/9

-2/9

1/9

Hence T₂^-1(x,y,z) = (x/9+4y/9-2z/9,-2x/9+y/9+4z/9,4x/9-2y/9+z/9).

(c ) If T₁(x,y) = (5,-4), then we have 2x+11y = 5 and 10x+56y = -4. On solving these equations, we get x = 162 and y =-29.

(d) If T₂(x,y,z) = (6,-5,-1), then we have x+2z = 6, 2x+y = -5 and 2y+z = -1. On solving these equations, we get x = -4/3, y =-7/3 and z =11/3

Note: We have reduced the augmented matrix of the linear system to its RREF for solving the equations.

2	11
10	56