Suppose that T is a linear transformation such that

Question

Suppose that  is a linear transformation such that

T(1,-2)=(8,-4) , T(-2,-1)=(-1,8)

Write T  as a matrix transformation.

the linear transformation T is given by 

[ , ]

[ , ]

Answer 1

Set this up as 2 systems of 2 linear equations in 2 unknowns each. From T(1,-2)=(8,-4) get

1a₁₁ - 2a₁₂ = 8

1a₂₁ - 2a₂₂ = -4

From T(-2,-1)=(-1,8) get

-2a₁₁ - a₁₂ = -1

-2a₂₁ - a₂₂ = 8

The 2 systems you need to solve are

a₁₁ - 2a₁₂ = 8

-2a₁₁ - a₁₂ = -1

and

a₂₁ - 2a₂₂ = -4

-2a₂₁ - a₂₂ = 8

Solve both systems by whatever method you prefer. Using Cramer's rule, my answer for the matrix elements is

a₁₁ = 2

a₁₂ = -3

a₂₁ = -4

a₂₂ = 0

Test the answer and see that it works.