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**Unformatted text preview: **Practice Problems 1/18/06 (1) A m n matrix A is called u pper triangular if all the entries lying below the diagonal entries are zero. That is, A ij = 0, whenever i > j . Prove that the upper triangular matrices form a subspace of M m n ( F ) . Solution: Clearly, the m n null matrix 0 is upper triangular. For two m n upper triangular matrices A = ( a ij ) , B = ( b ij ) , A + B = ( a ij + b ij ) is also upper triangular since a ij + b ij = 0 , whenever i > j. Similarly, kA is also UT for any k F . (2) Let S be a nonempty set and F a field. Let C ( S,F ) denote the set of all functions f F ( S,F ) such that f ( s ) = 0 for all but a finite number of elements of S . Prove that C ( S,F ) is a subspace of F ( S,F ). Solution: For f ( s ) , the zero element of F ( S,F ) is clearly in C ( S,F ) . Now, let f ( s i ) = c i , for c i 6 = 0 ,s i S,i = 1 ,....,n, and g ( t i ) = d i , for d i 6 = 0 ,t i S,i = 1 ,..,m and for all other s S let f ( s ) = 0 ,g ( s ) = 0. Now, ( f + g )( s ) = 0 for all...

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