Lect10 - Chapter 3. Vector Spaces Math1111 Vector Spaces...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a ∈ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T ∈ W , (ii) α ( a 1 ) T = ( α a 1 ) T for any scalar α and ( a 1 ) T ∈ W . Is W a vector space with respect to these two operations? Yes Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a ∈ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T ∈ W , (ii) α ( a 1 ) T = ( α a 1 ) T for any scalar α and ( a 1 ) T ∈ W . Is W a vector space with respect to these two operations? Yes Theorem 3.1.1 Let V be a vector space and x ∈ V . Then (i) x = , (ii) x + y = implies y =- x (i.e. additive inverse is unique) , (ii)’ Zero element is unique, (iii) (- 1 ) x =- x . (i.e. The scalar multiple (- 1 ) x is the additive inverse of x .) Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . The set R n with the usual addition and scalar multiplication is a vector space. Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . The set R n with the usual addition and scalar multiplication is a vector space. Example . Let P n be the set of all polynomials of degree ≤ n . Under the usual operations, P n is a vector space. ( Addition: Let p , q ∈ P n . Define p + q as ( p + q ) ( x ) = p ( x ) + q ( x ) . Scalar multiplication: Let α ∈ R & p ∈ P n . Define α p as ( α p ) ( x ) = α p ( x ) . ) Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . The set R n with the usual addition and scalar multiplication is a vector space. Example . Let P n be the set of all polynomials of degree ≤ n . Under the usual operations, P n is a vector space. ( Addition: Let p , q ∈ P n . Define p + q as ( p + q ) ( x ) = p ( x ) + q ( x ) . Scalar multiplication: Let α ∈ R & p ∈ P n . Define α p as ( α p ) ( x ) = α p ( x ) . ) Example . Let C [ a , b ] be the set of all real-valued functions on the interval [ a , b ] . Under the usual operations, C [ a , b ] is a vector space. Chapter 3. Vector Spaces Math1111 Subspaces Exercise Exercise Let V = { f : f is differentiable infinitely many times on...
View Full Document

This document was uploaded on 05/04/2011.

Page1 / 25

Lect10 - Chapter 3. Vector Spaces Math1111 Vector Spaces...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online