College Algebra Exam Review 158

College Algebra Exam Review 158 - 168 3. PRODUCTS OF GROUPS...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
168 3. PRODUCTS OF GROUPS Proof. It follows from Propostion 3.3.25 that any basis of a finite di- mensional vector space is finite. If a finite dimensional vector space has two bases X and Y , then j Y j ± j X j , since Y is linearly independent and X is spanning. But, reversing the roles of X and Y , we also have j Y j ± j X j . n Definition 3.3.27. The unique cardinality of a basis of a finite– dimensional vector space V is called the dimension of V and denoted dim .V / . If V is infinite–dimensional, we write dim .V / D 1 . Corollary 3.3.28. Let W be a subspace of a finite dimensional vector space V . (a) Any linearly independent subset of W is contained in a basis of W . (b) W is finite dimensional, and dim .W / ± dim .V / . (c) Any basis of W is contained in a basis of V . Proof. Let Y be a linearly independent subset of W . Since no linearly independent subset of W has more than dim .V / elements, by Proposition 3.3.25 , Y is contained in linearly independent set B that is maximal among
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online