1.6+2.1a

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 } ).
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online