Solve with complete steps and equations used Problem 2 Let T

Problem 2. Let T R3 R3 be a linear transformation such that and T (a) Find the matrix A of T, and give the explicit formula for T.

Solution

2.(a) We know that the columns of the standard matrix A of T are T(e₁),T( e₂) and T( e₃ ) where e₁ = (1,0,0), e₂ = (01,0)^T and e₃ = (0,0,1)^T. In order to determine these columns of A, we will reduce to RREF, as under, the matrix B =

1

2

1

1

0

0

2

5

4

0

1

0

1

1

0

0

0

1

Add -2 times the 1st row to the 2nd row

Add 1 times the 2nd row to the 3rd row

Add -1 times the 1st row to the 3rd row

Add -2 times the 3rd row to the 2nd row

Add -1 times the 3rd row to the 1st row

Add -2 times the 2nd row to the 1st row

Then the RREF of B is

1

0

0

-4

1

3

0

1

0

4

-1

-2

0

0

1

-3

1

1

This implies that e₁ = -4(1,2,1)^T+4(2,5,1)^T-3(1,4,0)^T, e₂ = (1,2,1)^T-(2,5,1)^T+(1,4,0)^T, and e₃ = 3(1,2,1)^T                    -2(2,5,1)^T+(1,4,0)^T. Now, since T is a linear transformation, have T(e₁) =-4T(1,2,1)^T+4T(2,5,1)^T-3T(1,4,0)^T = -4(2,0,1)^T +4(-1,1,0)^T -3(0,-2,2)^T = ( -8,0,-4)^T +(-4,4,0)^T+(0,6,-6)^T = ( -12, 10,-10)^T ,T( e₂) = T(1,2,1)^T-T(2,5,1)^T+ T(1,4,0)^T = (2,0,1)^T- (-1,1,0)^T+(0,-2,2)^T = ( 3,-3,3)^T and T( e₃ ) = 3T(1,2,1)^T   -2T(2,5,1)^T+ T(1,4,0)^T= 3(2,0,1)^T-2(-1,1,0)^T+(0,-2,2)^T = (6,0,3)^T+(2,-2,0)^T+ (0,-2,2)^T = ( 8,-4,5)^T. Then A =

-12

3

8

10

-3

-4

-10

3

5

Also T(X) = AX for any vector X = (x₁,x₂,x₃)^T.

(b) Having determined A, we have T9-2x) = -2T(x) = -2Ax =

-24x₁+6x₂+16x₃

20x₁-6x₂-8x₃

-20x₁+6x₂+10x₃

Then S(x) =

(-24x₁+6x₂+16x₃)+(2x₁+x₃)

(20x₁-6x₂-8x₃)+(x₂-x₃)

(-20x₁+6x₂+10x₃)+(x₁+x₂)

= Bx =

(-22x₁+6x₂+17x₃)

(20x₁-5x₂-9x₃)

(-19x₁+7x₂+10x₃)

so that B=

-22

6

17

20

-5

-9

-19

7

10

1	2	1	1	0	0
2	5	4	0	1	0
1	1	0	0	0	1