{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

10w_math115a_hw2_solutions

# 10w_math115a_hw2_solutions - Math 115A Homework 2 Solutions...

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

Math 115A Homework # 2 Solutions Prof. Yehuda Shalom TA: Darren Creutz Date Due: 21 Jan 2010 1.4.1 (a) T: write 0 = 0 x for any x in the set (b) F: 0 is in span ( ) (c) T: Thm 1.5 implies that span ( S ) W for any subspace W containing S so span ( S ) is contained in the intersection of all such subspaces; span ( S ) itself is a subspace containing S so the intersection of all such subspaces is contained in span ( S ). (d) F: any nonzero constant is ok (e) T: the new equation will contain the information from the old one (f) F: 0 = 1 has no solutions 1.4.6 Show that the vectors (1 , 1 , 0), (1 , 0 , 1) and (0 , 1 , 1) generate F 3 . Proof. Let v F 3 . Then v = ( a, b, c ) for some a, b, c F . Set r = 1 2 ( a + b - c ) s = 1 2 ( a - b + c ) t = 1 2 ( - a + b + c ) Then r (1 , 1 , 0) + s (1 , 0 , 1) + t (0 , 1 , 1) = ( r + s, r + t, s + t ) = 1 2 ( a + b - c + a - b + c, a + b - c - a + b + c, a - b + c - = ( a, b, c ) making use of the axioms of vector spaces (commutativity and definition of scalar multiplication). Hence v is a linear combination of the given three vectors. So span ( { (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) } ) = F 3 1.4.12 Show that a subset W of a vector space V is a subspace if and only if W =

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}