princ_techniques_draft

princ_techniques_draft - Chapter 1 Linear Spaces 1.1 Vector...

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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....
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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.

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princ_techniques_draft - Chapter 1 Linear Spaces 1.1 Vector...

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