Finding the kernel and image

Let A=
[1 1 1 1]
[0 1 2 3]
[0 2 -1 5]
[0 3 -4 7]
Find bases for ker(A) and Im(A). Are ker(A) and im(A) isomorphic? Why or why not?

First, row reduce A:
[1 0 0 -9/5]
[0 1 0 13/5]
[0 0 1 1/5]
[0 0 0 0].

Note we have one free variable. Solving for it (and rescaling the vector by 5) yields
ker(A) = {(9, -13, -1, 5)^t}, which has dimension 1.

Regarding Im(A), the pivot columns above correspond to linearly independent columns in the original matrix A: Im(A) = span{(1,0,0,0)^t, (1,1,2,3)^t, (1,2,-1,-4)^t}, which has dimension 3.

Since the kernel and image have different dimensions, they can't be isomorphic.

I hope this helps!