Lect12

# Lect12 - Chapter 3 Vector Spaces Math1111 Vector Spaces...

This preview shows pages 1–10. Sign up to view the full content.

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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 3. Vector Spaces Math1111 Vector Spaces Example Example . Show that S = ï£« ï£¬ ï£­ a d ï£¶ ï£· ï£¸ : a , d âˆˆ R with the usual matrix addition and scalar multipication is a vector space. Chapter 3. Vector Spaces Math1111 Vector Spaces Example Example . Show that S = ï£« ï£¬ ï£­ a d ï£¶ ï£· ï£¸ : a , d âˆˆ R with the usual matrix addition and scalar multipication is a vector space. Example . Let W = ( a 1 ) T : a is a real number . Is W with matrix addition and scalar multiplication a vector space? Chapter 3. Vector Spaces Math1111 Vector Spaces Example Example . Show that S = ï£« ï£¬ ï£­ a d ï£¶ ï£· ï£¸ : a , d âˆˆ R with the usual matrix addition and scalar multipication is a vector space. Example . Let W = ( a 1 ) T : a is a real number . Is W with matrix addition and scalar multiplication a vector space? NO Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a âˆˆ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T âˆˆ W , (ii) Î± ( a 1 ) T = ( Î± a 1 ) T for any scalar Î± and ( a 1 ) T âˆˆ W . Is W a vector space with respect to these two operations? Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a âˆˆ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T âˆˆ W , (ii) Î± ( a 1 ) T = ( Î± a 1 ) T for any scalar Î± and ( a 1 ) T âˆˆ 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 âˆˆ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T âˆˆ W , (ii) Î± ( a 1 ) T = ( Î± a 1 ) T for any scalar Î± and ( a 1 ) T âˆˆ 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 âˆˆ V . Then (i) x = , (ii) x + y = implies y =- x (i.e. additive inverse 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 Homework 2 Reading Textbook - p.121 Homework 2 Chapter 3 - Section 1 Exercises: Qn. 7, 8, 9, 10, 11, 12, 13, 14, 16. Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations. Chapter 3. Vector Spaces Math1111 Subspaces Motivation Example . Let S = ( x 1 x 2 ) T : x 1 = 2 x 2 } . Endow S with the addition and scalar multiplication of R 2 . Show that S is a vector space with respect to these two operations....
View Full Document

## This note was uploaded on 04/30/2010 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

### Page1 / 32

Lect12 - Chapter 3 Vector Spaces Math1111 Vector Spaces...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online