Please Help: Linear Algebra Question (linear transformations)

Question

 If T is a linear transformation such that

T([3, 2])=[21, 5, -2] and T([4, -1])=[28, -8, 1]

Then the matrix A that induces T is:

*I'm confused on how to even approach this problem*

Thank you

Answer 1

What you need to do is find the matrix elements of A. The matrix A must be a 3x2 matrix, so you must find 6 elements, which means you need 6 independent equations in the 6 unknown elements of A. The matrix equation

A(3,2) = (21,5,-2)

gives 3 equations

3A₁₁ + 2A₁₂ = 21

3A₂₁ + 2A₂₂ = 5

3A₃₁ + 2A₃₂ = -2.

Similarly, the matrix equation

A(4,-1) = (28,-8,1)

gives 3 equations

4A₁₁ - A₁₂ = 28

4A₂₁ - A₂₂ = -8

4A₃₁ - A₃₂ = 1.

You now have 6 equations in the 6 unknowns A₁₁,...,A_32. Solve for the A₁₁,...,A₃₂.