I suck

How do I find the matrix A for T(x) = Ax, where T: r2->r3 is a linear transformation with T([2, 1]^T) = [1, 0, 4]^T and T([1, 1]^T) = [0, 2, 2]^T ?

make a matrix with a bunch of vars in it, multiply by the vectors and solve.

Lin transforms are uniquely defined by what they do to a basis.  [2, 1]^T - [1, 1]^T = [1,0]^T smells awfully like a basis vector.  Once you have that, [1, 1]^T - [1,0]^T = [0,1]^T, which also smells like a basis vector.  Figure out what T does to these basis vectors, and you're done.

T([1,0]) = T([2,1] - [1,1]) = T([2,1]) - T([1,1]) = [1,0,4] - [0,2,2] = [1,-2,2]

T([0,1]) = T([1,1] - [1,0]) = T([1,1]) - T([1,0]) = [0,2,2] - [1,-2,2] = [-1,4,0]

So T is

[ 1, -1]
[-2,  4]
[ 2,  0]