hw03 - Math 110-Linear Algebra Fall 2009, Haiman Problem...

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Math 110—Linear Algebra Fall 2009, Haiman Problem Set 3 Due Monday, Sept. 21 at the beginning of lecture. 1. Show that if vectors v 1 ,...,v k in a vector space V have the properties that v 1 6 = 0, and each v i is not in the span of the preceding ones, then the vectors are linearly independent. Conversely, show that if v 1 ,...,v k is an ordered list of linearly independent vectors, then it has the above properties. 2. (a) Find a formula for the number Q ( n,k ) of (ordered) sequences ( v 1 ,v 2 ,...,v k ) of linearly independent vectors in V , where V is a vector space of dimension n over F 2 , and k n . [Hint: Use the previous problem and Problem Set 2, Problem 5.] (b) Prove that the number of k -dimensional subspaces of ( F 2 ) n is given by Q ( n,k ) /Q ( k,k ), for k n . (c) Calculate the number of 5-dimensional subspaces of ( F 2 ) 10 . 3. Let c 1 ,...,c n be distinct elements of a field F . Define the function E : P m ( F ) F n by E ( f ( x )) = ( f ( c 1 ) ,...,f ( c
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This note was uploaded on 08/16/2010 for the course MATH 110 taught by Professor Gurevitch during the Fall '08 term at Berkeley.

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