Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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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