Sorry for the lack of clarification. I had not meet questions like this before, so I wrote these two ideas to give it a try. I believed the second one should be more accurate as my functional analysis class did talk about why these functions

form a basis.

The image of A comes from finding a basis of it. I start by first finding a basis of L2, which are those simple functions. Then I check the image of those basis functions under A and those would form a basis of the image.