solution4 - Let V be the space of 2 2 matrices over F ....

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Homework assignment 4 pp. 49 Exercise 8. Let V be the space of 2 × 2 matrices over F . Find a basis { A 1 ,A 2 ,A 3 ,A 4 } for V such that A 2 j = A j for each j . Solution If we start with the canonical basis for V , namely B = B 1 = ± 1 0 0 0 ,B 2 = ± 0 1 0 0 ,B 3 = ± 0 0 1 0 ,B 4 = ± 0 0 0 1 ¶² we notice that the first and the last elements satisfy the required condition. Therefore we only need to find other two matrices, such that the four matrices generate V . Let A = A 1 = ± 1 0 0 0 ,A 2 = ± 1 1 0 0 ,A 3 = ± 0 0 1 1 ,A 4 = ± 0 0 0 1 ¶² Notice that A 2 j = A j and A 2 - A 1 = B 2 and A 3 - A 1 = B 3 . Therefore A is a basis for V Bonus exercise 14. Let V be the set of real numbers. Regard V as a vector space over the field of rational numbers, with the usual operations. Prove that this vector space is not finite-dimensional. Solution: By contradiction. Suppose that V is finite dimensional, this implies that V is a countable set (see the lemma below), but the set of real numbers is not countable! Lemma 1. A finite dimensional vector space V over the rational numbers is a countable set. Proof. By induction on n = the number of elements in a basis for V . Base: n = 1 Let { α } be a basis for V . Let { a 1 ,a 2 ,...,a n ,... } be an enumeration of the rational numbers, then any element of V may be written as a i α . That is, V ⊂ { a 1 α,a 2 α,. ..,a n α,. .. } (actually the two sets are equal, but we don’t need that fact). Therefore V is countable. Inductive Step: Suppose that V is countable if its dimension is less than or equal to n . We will prove then that if the dimension of V is n + 1 then it is a countable set . Let { β 1 ,...,β n } be a basis for V . By the induction hypothesis we know that the subspace W generated by { β 1 ,...,β n } is a countable set. Let 1
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{ w 1 ,w 2 ,...,w n ,... } be an enumeration for W . Consider the following infinite array: ( w 1 + a 1 α ) 1 ( w 1 + a 2 α ) 2 v n n n n n n n n n n n n ( w 1 + a 3 α ) 4 v n n n n n n n n n n n n ... ( w 1 + a n α ) y s s s s s s s s s s s s ... ( w 2 + a 1 α ) 3 ( w 2 + a 2 α ) 5 v n n n n n n n n n n n n ( w 2 + a 3 α ) 8 v n n n n n n n n n n n n ... ( w 2 + a n α ) y s s s s s s s s s s s s ... ( w 3 + a 1 α ) 6 ( w 3 + a 2 α ) 9 w o o o o o o o o o o o o o o o ( w 3 + a 3 α ) 13 w n n n n n n n n n n n n n n n n ... ( w 3 + a n α ) z t t t t t t t t t t t t ... . . . . . . . . . . . . . . . ( w
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.

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solution4 - Let V be the space of 2 2 matrices over F ....

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