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: Practice Problems 2/13/06 (1) Let u and v be two distinct vectors of a vector space V . Let { u,v } be a basis for V and a,b nonzero scalars. Show that { u + v,au } and { au,bv } are also bases for V . (2) The set of all (3 3) matrices having trace equal to zero is a subspace W of M 3 3 . Find a basis for W . Solution: Note that any (3 3) matrix a b c , can be written as a linear combination of 0 1 0 0 0 0 0 0 0 , 0 0 1 0 0 0 0 0 0 , 0 0 0 1 0 0 0 0 0 , 0 0 0 0 0 1 0 0 0 , 0 0 0 0 0 0 1 0 0 , 0 0 0 0 0 0 0 1 0 , 1 0 1 0 0 0 , 0 0 1 0 0 0 1 . This is a linearly independent, generating set. Hence the dimension of W is 8. (3) The set of all (3 3) upper triangular matrices is a subspace W of M 3 3 . Find a basis for W . (4) Let V be a vector space having dimension n , and let S be a subset of V that generates V . Prove that there is a subset of....
View
Full
Document
This note was uploaded on 02/01/2010 for the course MATH math115a taught by Professor Shalom during the Spring '10 term at UCLA.
 Spring '10
 SHALOM
 Linear Algebra, Algebra, Vectors, Matrices, Scalar, Vector Space

Click to edit the document details