Let T, U: V > W be linear transformations. (a) Prove that R(T + U) C R(T) + R(U). (See

Chapter 3, Problem 14

(choose chapter or problem)

Let T, U: V > W be linear transformations. (a) Prove that R(T + U) C R(T) + R(U). (See the definition of the sum of subsets of a vector space on page 22.) (b) Prove that if W is finite-dimensional, then rank(T+U) < rank(T) + rank(U). (c) Deduce from (b) that rank(.A + B) < rank(A) + rank(5) for any mxn matrices A and B.

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