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Linear mapping/Dimension formula/Fact/Proof

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Proof

Set . Let denote the kernel of the mapping and let denote its dimension (). Let

be a basis of . Due to fact, there exist vectors

such that

is a basis of . We claim that

is a basis of the image. Let be an element of the image . Then there exists a vector such that . We can write with the basis as

Then we have

which means that is a linear combination in terms of the . In order to prove that the family , , is linearly independent, let a representation of zero be given,

Then

Therefore, belongs to the kernel of the mapping. Hence, we can write

Since this is altogether a basis of , we can infer that all coefficients are , in particular, .