hw2 - (b) Show that the subset S ( n ) M ( n ) of symmetric...

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Math 136A, Spring 2008 Problem Set #2 (due 4/11/2008) (1) Let V denote the set of infinitely differentiable functions on the interval [0 , 1]. It is a vector space under the oprations of addition of functions and multiplication by a real number. Let W = { f V : f 00 + pf 0 + qf = 0 } for p V and q V . (a) Prove that W is a subspace of V . (b) What does the existence and uniqueness theorem for ordinary differential equa- tions tell you about the dimension of W ? (2) Let M ( n ) be the set of n × n matrices. (a) Show that M ( n ) is a vector space, where addition is addition of matrics and multiplication by scalars is defined as on page 45 of Lang.
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Unformatted text preview: (b) Show that the subset S ( n ) M ( n ) of symmetric matrices is a subspace. What is its dimension? (c) Show that the subset A ( n ) M ( n ) of skew-symmetric matrices is a subspace. What is its dimension? (d) Show that M ( n ) = S ( n ) + A ( n ) and S ( n ) A ( n ) = O . (3) In problem (1) above you showed that the space of innitely dierentiable functions on the unit interval is a vector space. Show that V is not nite dimensional. Hint: Consider the functions f k ( x ) = x k , k = 0 , 1 , 2 , 3 ,... . 1...
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This note was uploaded on 10/12/2009 for the course MATH 136 taught by Professor Staff during the Spring '08 term at University of Washington.

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