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ECEN601_hw5_pfister

# ECEN601_hw5_pfister - ECEN 601 Assignment 5 Problems 1 Let...

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ECEN 601: Assignment 5 Problems: 1. Let W 1 and W 2 be subspaces of a vector space V such that W 1 + W 2 = V and W 1 W 2 = { 0 } . Prove that for each vector α in V there are unique vectors α 1 in W 1 and α 2 in W 2 such that α = α 1 + α 2 . Note: If S 1 ,S 2 ,...,S k are subsets of a vector space V , the set of all sums α 1 + α 2 + · · · + α k or vectors α i in S i is called the sum of the subsets S 1 ,S 2 ,...,S k and is denoted by S 1 + S 2 + · · · + S k of by k summationdisplay i =1 S i . 2. Let V be the set of all complex-valued functions f on the real line such that (for all t in R ) f ( t ) = f ( t ) . The bar denotes complex conjugation. Show that V , with the operations ( f + g )( t ) = f ( t ) + g ( t ) ( cf )( t ) = cf ( t ) is a vector space over the field of real numbers. Give an example of a function in V which is not real-valued. 3. Let F be a field and let n be a positive integer ( n 2). Let V be the vector space of all n × n matrices over F . Which of the following sets of matrices A in V are subspaces of V ?

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