Linear Transformation ( T : R2 --> R4)

Question

Let T : R2 --> R4 be a linear transformation such that,

T[ 1 1]^T = [ -1 1 2 3]^T , T[-1 1]^T = [2 0 2 3]^T

Find T[5 3]^T

Answer 1

[5 3] = a[1 1] + b[-1 1]

a - b = 5

a + b = 3

2a = 8, a = 4

b = - 1

So T[5,3] = T[4[1 1] - [- 1 1]] = 4T[1 1] - T[-1 1] = 4[-1 1 2 3] - [2 0 2 3] = [- 6 4 6 9]