1.62.1 - Homework 3 - Solutions 1.6 Problem 1 1. FALSE. The...

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Homework 3 - Solutions 1.6 Problem 1 1. FALSE. The zero vector can be taken as a (trivial) basis. 2. TRUE. See Theorem 1.9. 3. FALSE. See Example 5 in Section 1.6. 4. FALSE. See Example 6, and the standard basis of R 3 . 5. TRUE. See Corollary 1 on page 46. 6. FALSE. The dimension is n +1. 7. FALSE. The dimension is mn . 8. TRUE. See Theorem 1.10. 9. TRUE. See Corlooary 2a on page 47. 10. TRUE. See Theorem 1.11. 11. TRUE. { 0 } and V itself are the subspaces in question. 12. TRUE. (See Corollary 2 on page 47). 1.6 Problem 2a We try to express (0 , 4 , 3) as a linear combination of the ±rst two vectors. If there exist constants c 1 and c 2 such that c 1 (1 , 0 , 1) + c 2 (2 , 5 , 1) = (0 , 4 , 3) then c 1 +2 c 2 =0 5 c 2 = 4 c 1 + c 2 =3 Solving the last two equations, we get c 2 = 4 5 and c 1 = 19 5 But these values do not satisfy the ±rst equation. Therefore, the three given vectors are independent, and hence form a basis of R 3 .
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1.6 Problem 9 We have to fnd constants c 1 ,c 2 3 4 , such that c 1 u 1 + c 2 u 2 + c 3 u 3 + c 4 u 4 = a 1 a 2 a 3 a 4 Solving this systemn oF equations (starting From c 4 and working backwards), we get ( a 1 ,a 2 3 4 )=( a 1 a 2 2 a 3 3 a 4 4 ) 1.6 Problem 10a P ( x )= 4 x 2 x +8 (See the end oF section 1.6 For a similar example). 1.6 Problem 17 Let V ij be defned as Follows, For 1 i<j n :The ij and ji entries oF V ij are 1 and -1 respectively. All the remaining entries are zero. Then the set oF all matrices { V ij } Forms a basis oF the space oF skew-symmetric matrices oF order n × n . The total number oF such matrices is n ( n 1) 2 . ±or each fxed i ,thereare n i values oF j ( j = i +1 ,...,n ). ThereFore, adding over all possible values oF i ,weget dim( W n 1) + ( n 2) + ··· +1= n ( n 1) 2 1.6 Problem 22 Claim 1. W 1 W 2 = dim( W 1 W 2 )=dim( W 1 ) Proof. IF W 1 W 2 ,then W 1 W 2 = W 1 . ThereFore dim( W 1 W 2 )=d im( W 1 ) Claim 2. dim( W 1 W 2 W 1 W 1 W 2 Proof. W 1 W 2 is a subspace oF W 1
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.

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1.62.1 - Homework 3 - Solutions 1.6 Problem 1 1. FALSE. The...

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