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I was trying to prove the following result :

for a linear mapping L: V --> W

dimension of a domain V = dimension of Im (L) + dimension of kernel (L)

So, my doubt actually is that do we really need a separate basis for the kernel ?

Theoretically, the kernel is a subspace of the domain V . So, the basis for V can be used to express any vector in the kernel .

If we really need it, i think this might be the reason ( please check whether i am right or not )

The kernel directs the mapping to the zero vector. We are not interested in this trivial mapping. So, even though the basis vectors of V can clearly describe the vectors in kernel, we keep the vectors in kernel aside and are interested in only those vectors in V which do not produce the zero vector.

Then, since T(ki) = 0 where ki represents any vector in the kernel , ki too ought to be a part of the solution for any mapping. Hence, a separate basis which only can express any vector in the kernel is needed

Am i thinking on the right line ?

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# What is the need for a separate basis for the kernel ? Please have a look at my expln

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