Let V, W be finite-dimensional vector spaes over C.(a) If T ≤ L(V, W), prove the Rank-Nullity Theorem with T.(b) If V = Mnxn is the space of n - n matrices over C and A, B E Mnxn, proverank(AB) ≤ min{rank(A), rank(B)},rank(A + B) ≤ rank(A) + rank(B).

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Let V, W be finite-dimensional vector spaes over C. (a) If T ≤ L(V, W), prove the Rank-Nullity Theorem with T. (b) If V = Mnxn is the space of n x n matrices over C and A, B E Mnxn, prove rank(AB) ≤ min{rank(A), rank(B)}, rank(A + B) ≤ rank(A) + rank(B).

Let V, W be finite-dimensional vector spaes over C. (a) If T ≤ L(V, W), prove the Rank-Nullity Theorem with T. (b) If V = Mnxn is the space of n x n matrices over C and A, B E Mnxn, prove rank(AB) ≤ min{rank(A), rank(B)}, rank(A + B) ≤ rank(A) + rank(B).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let V, W be finite-dimensional vector spaes over C.
(a) If T ≤ L(V, W), prove the Rank-Nullity Theorem with T.
(b) If V = Mnxn is the space of n - n matrices over C and A, B E Mnxn, prove
rank(AB) ≤ min{rank(A), rank(B)},
rank(A + B) ≤ rank(A) + rank(B).

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