Vector Spaces Sets Closed Under Operations Defn. A set S is closed - PowerPoint PPT Presentation

Vector Spaces

Sets Closed Under Operations Defn. A set S is closed under some opera- tion if applying that operation to elements of S always produces an element of S . For example, the integers are closed under ad- dition: adding two integers always produces an integer. The integers are also closed under sub- traction and multiplication, but not division. spaceONE: 2

Sets Closed Under Operations The set of positive real numbers is not closed under subtraction: for example, 2 − π is not pos- itive. The set is closed under addition, multipli- cation, division, and exponentiation. spaceONE: 3

A Span is Closed Fact. The set of solutions to the homogeneous equation A x = 0 is closed under both addition and scalar multiplication. In other word, if you add two solutions of the homogenous equation then the result is still a solution; similarly, if you scale a solution then the result is still a solution. spaceONE: 4

Vector Spaces Defn. A vector space is a collection of ob- jects (called vectors ) with operations addition and scalar multiplication that obey the “usual” vector laws. That is, addition and scalar multiplication be- have like they do for ordinary vectors. spaceONE: 5

Overview of The Axioms of a Vector Space ≻ the space is closed under addition and scalar multiplication ≻ addition is commutative (order doesn’t mat- ter), associative (brackets don’t matter), and has negation ; ≻ the 0 vector and 1 scalar behave as identi- ties ; and ≻ addition and scalar multiplication distribute (interact nicely). spaceONE: 6

Axioms of a Vector Space For all vectors u , v , and w in V and all (real) scalars c and d : 1) The sum u + v is in V 2) u + v = v + u 3) ( u + v ) + w = u + ( v + w ) 4) There is a vector 0 such that u + 0 = u 5) There is a vector − u such that u + ( − u ) = 0 spaceONE: 7

Axioms of a Vector Space Continued 6) The scalar multiple c u is in V 7) c ( u + v ) = c u + c v 8) ( c + d ) u = c u + d u 9) c ( d u ) = ( cd ) u 10) 1 u = u spaceONE: 8

Example R n R n , with addition and scalar multiplication as we’ve been doing, is a vector space. spaceONE: 9

Polynomials form Vector Space Defn. P n is the set of all polynomials of degree at most n ; P is the set of all polynomials. These are vector spaces. spaceONE: 10

Functions form Vector Space Defn. C [ t ] is the set of all continuous functions in variable t with domain R This is a vector space. One adds and scales functions just as in calculus. spaceONE: 11

Matrices form Vector Space Defn. M n is the set of all n × n matrices. This is a vector space. spaceONE: 12

Summary A set is closed under a operation if applying that operation to elements of the set always pro- duces an element of the set. A vector space is a collection of objects with ad- dition and scalar multiplication that obey the “usual” vector laws: it is closed under addition and scalar multiplication; addition is commuta- tive, associative and has negation; the 0 vector and 1 scalar act as identities; and addition and scalar multiplication distribute. spaceONE: 13

Summary (cont) Examples include R n ; P n the set of all polynomi- als of degree at most n ; C [ t ] the set of all contin- uous functions in variable t ; and M n the set of all n × n matrices. spaceONE: 14