Linear Transformation and Standard Matrix

Question

Let T: R²-> R² be the linear transformation with the property:

T(1, 0) = (4, -1) and T(2, 1) = (6, 1)

Find [T], the standard matrix of T

Answer 1

Let T is 2×2 matrix with columns (a, c) and (b, d). Then T(1, 0) = (a, c) = (4, -1). From here a = 4, c = -1.

Then T(2,1) = (2a + b, 2c + d) = (8 + b, -2 + d) = (6, 1). From here 8 + b = 6 and -2 + d = 1. Because

b = -2 and d = 1 + 2 = 3.

Answer: matrix T of this linear transformation has 1st column (4, -1) and 2nd column (-2, 3)