Bourbaki

Exercise 11, § 7

Algebra, II, 403

Let E, F be two vector spaces over a field K and u:E→Fu : E \to F a linear mapping. Show that for every vector subspace V of F,

dim⁡(u−1(V))=dim⁡(V∩Im(u))+dim⁡(Ker(u)). \dim(u^{-1}(V)) = \dim(V \cap \mathrm{Im}(u)) + \dim(\mathrm{Ker}(u)).

Solution

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Transcribed from the printed volume. Not translated.