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**0**}.

Now it seems to be like this would be false because if the kernel, aka nullspace, is zero then the matrix will have a non-zero range. Also, the range and kernel have dimensions in different spaces, and they will not have the same bases. I can see how this would work for a function, for example f(x)= x^2 = 0, and then show that f(1) =/= 0.

I think it ultimately comes down to showing that:

Ax = 0

Ax = b = x

Am I on the right track?