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What makes it the most fundamental?

It's basically the matrix form of dim(domain(f)) = dim(ker(f)) + dim(im(f)), or domain(f)/ker(f) ≅ im(f) with f (inducing) the isomorphism.

But the image and kernel are not the only two properties of a matrix.

Sure, but from an algebraic (categorical) perspective, taking kernels, co-kernels and images are the most fundamental operations you can think of.

The question is: How do you compute them in an effective (avoiding full SVD) and stable way?