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

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Oct 22, 2023 278 likes •420 views

Vector Spaces Sets Closed Under Operations Defn. A set S is closed under some operation if applying that operation to elements of S always produces an element of S . For example, the integers are closed under addition: adding two integers

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Vector Spaces

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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.

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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.

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A Span is Closed

Fact. The set of solutions to the homogeneous equation Ax = 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.

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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.

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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).

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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

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Axioms of a Vector Space Continued

6) The scalar multiple cu is in V 7) c(u + v) = cu + cv 8) (c + d)u = cu + du 9) c(du) = (cd)u 10) 1u = u

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Example Rn

Rn, with addition and scalar multiplication as we’ve been doing, is a vector space.

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Polynomials form Vector Space

Defn. Pn is the set of all polynomials of degree at most n; P is the set of all polynomials. These are vector spaces.

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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.

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Matrices form Vector Space

Defn. Mn is the set of all n × n matrices. This is a vector space.

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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.

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Summary (cont)

Examples include Rn; Pn the set of all polynomi- als of degree at most n; C[t] the set of all contin- uous functions in variable t; and Mn the set of all n × n matrices.

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