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Unformatted text preview: 1) Give an example of an infinite dimensional vector space V over R , and linear operators T and S such that: A) S is onto, but not one-to-one. B) T is one-to-one, but not onto. 2) State true or false and justify: if V is a finite-dimensional vector space and W 1 and W 2 are subspaces of V such that V = W 1 ⊕ W 2 , then for any subspace W of V we have W = ( W ∩ W 1 ) ⊕ ( W ∩ W 2 ). False. Example: < (1 , 1) > ⊕ < (0 , 1) > = R , but for W = (1 , 0) we have ( < (1 , 0) > ∩ < (1 , 1) > ) ⊕ ( < (1 , 0) > ∩ < (1 , 1) > ). 3) Let F be a field, take m,n ∈ Z + and let A ∈ F m × n be an m × n matrix. A) Define “row space of A ” B) Define “col space of A ” C) Prove that the dimension of the row space of A is equal to the dimension of the column space of A . 4) Let D be a principal ideal domain, let n ∈ Z + and let D ( n ) denote a free D-module of rank n . A) If L is a submodule of D ( n ) , prove that L is a free D-module of rank m ≤ n . B) If L is a proper submodule of D ( n ) , prove or disprove that the rank of L must be less than n . 5) Let D be a principal ideal domain and V and W denote free D-modules of rank 5 and 4 respectively. Assume that φ : V → W is a D-module homomorphism, and that B =...
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This document was uploaded on 01/25/2012.
- Spring '09
- Vector Space