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Assume that the linear transformation is known as a matrix.

I am aware of algorithms (by elementary row operations etc.) which calculate the rank of a matrix and hence the dimension of the kernel of the matrix.

But I would like to know of some non-algorithmic way of doing the same (if it exists). Like if there is a closed form expression which gives the kernel dimension in terms of the matrix entries? I am not sure how to make it precise..but I would like to have a method which can be used as a step in a proof which depends on the kernel dimension. An algorithmic process to calculate the same is not generally amenable to a mathematical proof.

I would eventually be interested in knowing the difference between the kernel dimension and the image dimension of a linear transformation. Hence telling me a method to calculate just the above difference would also be very helpful.

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$\dim(\mathrm{ker}(T)) - \dim(\mathrm{Im}(T)) = \dim(V) -2\dim(\mathrm{Im}(T))$. –  Arturo Magidin Mar 31 '11 at 20:43
    
If your proof depends on the dimension of the kernel, can you not simply leave the dimension indicated? You know the dimension exists; or perhaps you can do induction on the dimension of the kernel. –  Arturo Magidin Mar 31 '11 at 20:46
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