Linear Algebra - Transformation Matrices BASIC help. URGENT! Find T(x)?

My professor did a problem on the board, but he is extremely bad at teaching. He does the problem, but doesn't tell you how or why he did certain steps.

Here is the problem: (pretend [x,y] means that x is on top and y is on bottom in a column.(idk how to represent column on here)

If T[1,1] = [2,3] and T [1,-1] = [5,-1] find T(x).

What he did:

e1 = [1,0] = 1/2([1,1] +[1,-1]) = T(e1) =[7/2.1]

e2 = [0,1] = 1/2([1,1] - [1,-1]) = T(e2) = [-3/2,2]

A = [ 7/2,1 : -3/2,2]

T(x) = Ax

I understand all the basic math, but I have no clue how he got e1 , e2??? Did he just choose random numbers? Are they always [1,0] and [0,1]? Please help. I am clueless where he came up with these matrices.