princ_techniques_draft

# princ_techniques_draft - Chapter 1 Linear Spaces 1.1 Vector...

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

Unformatted text preview: Chapter 1 Linear Spaces 1.1 Vector Spaces Definition 1.1 (Vector Space) . A set V of elements v 1 ,v 2 ,... (called vectors) is a vector space over a field F if there is a mapping (called “addition”) ⊕ : V ×V → V such that for any u , v and w in V : 1. ( u ⊕ v ) ⊕ w = u ⊕ ( v ⊕ w ) (associative law), 2. u ⊕ v = v ⊕ u (commutative law), 3. there is a null element in V , called , such that v ⊕ 0 = 0 ⊕ v = v , 4. for every vector, there is an (additive) inverse called- v such that v ⊕ (- v ) = 0 , and there is a mapping (called “scalar multiplication”) : F × V → V such that for any c and d in F : 5. ( cd ) v = c ( d v ) (associative law), 6. ( c + d ) v = ( c v ) ⊕ ( d v ) (scalar) distributive law, 7. c ( u ⊕ v ) = ( c u ) ⊕ ( c v ) (vector) distributive law, and 8. there is a scalar identity in F , called 1 , such that 1 v = v . Since a lot of this notation may be new, we make a few comments here: • A scalar field F is a set of numbers that satisfy the common algebraic properties we are familiar with: associative rules, commutative rules, existence of additive and multiplicative identities and inverses, and distributive rules. Common examples are – R : the set of all real numbers, – C : the set of all complex numbers, and – Q : the set of all rational numbers. • The notation ⊕ : V×V → V implies that the addition of any two vectors in V produces a vector that must also belong to V . Thus, the vector space is closed under addition . This statement explicitly appears in many definitions of a vector space....
View Full Document

## This note was uploaded on 08/26/2010 for the course MATH MATH 5435 taught by Professor Brogghard during the Fall '10 term at Virginia Tech.

### Page1 / 3

princ_techniques_draft - Chapter 1 Linear Spaces 1.1 Vector...

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

View Full Document
Ask a homework question - tutors are online