{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows page 1. Sign up to view the full content.

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.
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (b) Show that the subset S ( n ) ⊂ M ( n ) of symmetric matrices is a subspace. What is it’s dimension? (c) Show that the subset A ( n ) ⊂ M ( n ) of skew-symmetric matrices is a subspace. What is it’s 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 inﬁnitely diﬀerentiable functions on the unit interval is a vector space. Show that V is not ﬁnite dimensional. Hint: Consider the functions f k ( x ) = x k , k = 0 , 1 , 2 , 3 ,... . 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online