Let f ∈ L (U, V ), g ∈ L (V, W) and let U, V be finite-dimensional. Prove that(a) dim im(g ◦ f) = dim im f − dim(ker g ∩ im f) and so dim im(g ◦ f) ≤ min(dim im f, dim im g);(b) dim ker(g ◦ f) = dim ker f + dim(ker g ∩ im f) and so dim ker(g ◦ f) ≤ dim ker f + dim ker g.

Let f ∈ L (U, V ), g ∈ L (V, W) and let U, V be finite-dimensional. Prove that(a) dim im(g ◦ f) = dim im f − dim(ker g ∩ im f) and so dim im(g ◦ f) ≤ min(dim im f, dim im g);(b) dim ker(g ◦ f) = dim ker f + dim(ker g ∩ im f) and so dim ker(g ◦ f) ≤ dim ker f + dim ker g.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 28E: 28. Let where and are nonempty. Prove that has the property that for every subset of if...

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Let f ∈ L (U, V ), g ∈ L (V, W) and let U, V be finite-dimensional. Prove that
(a) dim im(g ◦ f) = dim im f − dim(ker g ∩ im f) and so dim im(g ◦ f) ≤ min(dim im f, dim im g);
(b) dim ker(g ◦ f) = dim ker f + dim(ker g ∩ im f) and so dim ker(g ◦ f) ≤ dim ker f + dim ker g.

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