Show that the linear map T LR 3 with Tx1 x2 x3 3x2 x1x2 2

Show that the linear map T ? L(R 3 ) with T(x1, x2, x3) = (3x2, x1?x2, 2x1+x2+x3) is invertible. Find the matrices of T, T^?1 with respect to the standard basis of R^3 .

Solution

T can be written in matrix form Tx = B

T(x1, x2, x3) = (3x2, x1?x2, 2x1+x2+x3)

Let T^-1((3x2, x1?x2, 2x1+x2+x3) = (x1,x2,x3)

Then there must exist a,b,c scalars such that

a(3x2),b(x1?x2),c(2x1+x2+x3) = (1,1,1)

3a-b = 1

b +2c =1

c =1

Substitute c value in second to get b =-1

a = 0

Hence there is an inverse matrix as (0, -1, 1) as T inverse..