554-finalexam

# 554-finalexam - Give an example of an innite dimensional...

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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 = { v 1 , · · · , v 5 } is an ordered basis of V and B 0 = { w 1 , · · · , w 4 } is an ordered basis of W .

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