Regarding "interpret the columns of the matrix as the x and y axes of the coordinate frame", does that mean (a, b) is a point, and (c, d) is a point, making col0 the x axis and col1 the y axis? If so, how come earlier in the lecture the professor said that we think about points as column vectors in this class (bc in this sense, we'd interpret (a, c) and (b, d) each as a point right?)
(a, c) is the point that (1, 0), the unit vector in x direction, maps to under this transform. Likewise with (b,d) and (0, 1). This slide basically says that in order to figure out the matrix for a linear transform you have to ask yourself 2 questions: Where do (1,0) and (0, 1) map to? The answers respectively are the first and second column of the matrix. You could interpret (a, b) and (c, d) as points but there significance is only that they are linear combinations for the first element and second element respectively for the output of this transform.