This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: AS3sol/MATH1111/YKL/0809 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 3 Suggested Solution 1. (a) No matter whether or not W is a subspace, W ⊂ Span( W ) since w ∈ W ⇒ 1 · w ∈ Span( W ). ”If part”: If W is a subspace, then any linear combination of elements of W belongs to W . i.e. Span( W ) ⊂ W . ∴ W = Span( W ). ”Only if” part: Clear by direct checking from definition. (b) As in part (a), we know S ⊂ Span( S ) and Span( S ) is a subspace. Want to show: S ⊂ W and W is a subspace ⇒ Span( S ) ⊂ W . This is also clear, since w 1 ,w 2 , ··· ,w r ∈ S ⇒ w 1 ,w 2 , ··· ,w r ∈ W . Thus, a 1 w 1 + a 2 w 2 + ··· + a r w r ∈ W because W is a subspace. ∴ Span( S ) ⊂ W . 2. (a) Since U + V is a subspace containing the sets U and V , by Qn 1 part (b), Span( U ∪ V ) ⊂ U + V . Let x ∈ U + V . Then x = u + v where u ∈ U and v ∈ V . So x is a linear combination of vectors in U ∪ V . i.e. U + V ⊂ Span( U ∩ V ). (b) No. Consider U = Span((1 1) T ), V = Span( e 1 ), W = Span( e 2 ), all of which are subspaces of R 2 . Then U + ( V ∩ W ) = U + { } = U but ( U + V ) ∩ ( U + W ) = R 2 ∩ R 2 = R 2 . Remark. U + ( V ∩ W ) ⊂ ( U + V ) ∩ ( U + W ) is always true. (c) No. Use the counterexample in (b). Now, U ∩ V = { } and U ∩ W = { } , but since V + W = R 2 , we have U ∩ ( V + W ) = U . 3. (a) dim X = 1 as { x } is a basis for X . Similarly, dim Y = 1. dim W = 2 as { u , v } is a basis for W . dim( X + W ) = 2 as X + W = { t + s : t ∈ X, s ∈ W } = { α x + β u + γ v : α,β,γ ∈ R } . However, x = 2 u + 2 v so X + W = Span( u , v ) where u , v are linearly independent. dim( Y + W ) = 3 as Y + W = Span( y , u , v ) and y , u , v are linearly independent....
View
Full
Document
This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.
 Fall '08
 forgot
 Math, Linear Algebra, Algebra

Click to edit the document details