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1.6+2.1a - 1 Math 110 Homework 3 Partial Solutions If you...

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1 Math 110 Homework 3 Partial Solutions If you have any questions about these solutions, or about any problem not solved, please ask via email or in office hours, etc. 1.6.23 In this case, by a previous homework problem, we already have that W 1 W 2 . (a) I claim that dim( W 1 )=dim( W 2 ) if and only if W 1 = W 2 if and only if v span( { v 1 , . . . , v k } ). The first if and only if is straightforward. For the second, suppose first that W 1 = W 2 . Then v W 1 . This proves that v is in the span of { v 1 , . . . , v k } . On the other hand, suppose that v span( { v 1 , . . . , v k } ). We will then show that W 2 W 1 and thus that they are equal. Let x W 2 . Then there exist scalars a 1 , . . . , a k , a k +1 F such that x = a 1 v 1 + · · · + a k v k + a k +1 v . But v span( { v 1 , . . . , v k } ), so there are b 1 , . . . , b k F such that v = b 1 v 1 + · · · + b k v k . Then x = a 1 v 1 + · · · + a k v k + a k +1 ( b 1 v 1 + · · · + b k v k is in span( { v 1 , . . . , v k } ). (b) If dim( W 1 ) = dim( W 2 ), then it must be that dim( W 1 ) + 1 = dim( W 2 ). Choose a basis β for W 1 from among the vectors { v 1 , . . . , v k } .
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