If T: R^2 to R^3 is a linear transformation satisfying T(1,1)=(8,3,1) and T(1,-1) = (4,1,5)

Question

what is the first, second and third component of T(3,1)?

Answer 1

First, express (3,1) as a linear combination of (1,1) and (1,-1):

 

     a(1,1) + b(1,-1) = (3,1)

 

     (a+b, a-b) = (3,1)      So, a+b = 3

                                           a-b =  1

 

                                             2a = 4....Therefore, a = 2 and b = 1

 

So, (3,1) = 2(1,1) + 1(1,-1)

 

T(3,1) = T[2(1,1) + 1(1,-1)] = 2T(1,1) + 1T(1,-1)

 

                                          = 2(8,3,1) + 1(4,1,5) = (20, 7, 7)