lecture12 - Rn Fill in the blanks u1, u2, , uk c1u1+ c2u2 +...

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

View Full Document Right Arrow Icon
Chapter 3 Vector spaces 1 Fill in the blanks R n c 1 u 1 + c 2 u 2 +… + c k u k u 1 , u 2 , …, u k span{ u 1 , u 2 , …, u k } 1. c 1 u 1 + c 2 u 2 +… + c k u k is called a linear combination of u 1 , u 2 , …, u k . 2. The set of all linear combinations of u 1 , u 2 , …, u k is called the linear span of u 1 , u 2 , …, u k . 3. The subset span{ u 1 , u 2 , …, u k } is called a subspace of R n . 4. If there is no “ redundant ”vectorin{ u 1 , u 2 , …, u k }, we say the set is linearly independent 5. If { u 1 , u 2 , …, u k } is linearly independent and span{ u 1 , u 2 , …, u k } = V, then { u 1 , u 2 , …, u k } is called a basis for V.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Chapter 3 Vector Spaces 2 Lecture 12 3.4 Bases 3.5 Dimensions
Background image of page 2
Lecture 12 Announcement 3 Announcement ± Workbin ² Tutorial 4 and Lab 2 solutions ² Exercise set 3 (Q1-16) solutions ² Practice Session 5 answers ± Mid-term Test next Wednesday More info during the break ± No Lab session next week
Background image of page 3

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

View Full DocumentRight Arrow Icon
Chapter 3 Vector spaces 4 Recall from previous lectures To Show: S = { u 1 , u 2 , …, u k } spans R n To Show: S = { u 1 , u 2 , …, u k } is lin. indep. c 1 u 1 + c 2 u 2 + ··· + c k u k = v v is any general vector in R n check whether the system is always consistent c 1 u 1 + c 2 u 2 + ··· + c k u k = 0 0 is the zero vector in R n check whether the system has non-trivial solution yes no spans does not span yes no lin.dep lin.indep Given that : S = { u 1 , u 2 , …, u k } is a subset of R n same as : span (S) = R n
Background image of page 4
Chapter 3 Vector spaces 5 Exercise (similar to Ex 3 Q18(b) ) Given u , v , w are linearly independent Are u + v , u + w , v + w linearly independent? Consider a ( u + v ) + b ( u + w ) + c ( v + w )= 0 (*) Does (*) have non-trivial coefficients for a, b, c ? Rewrite (*) : (a+b) u + (a+c) v + (b+c) w = 0 (**) (**) has only trivial coefficients for a+b, a+c, b+c a + b = 0 a + c = 0 b + c = 0 Solve: a = b = c = 0 So (*) has only trivial coefficients for a, b, c So u + v , u + w , v + w are linearly independent
Background image of page 5

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

View Full DocumentRight Arrow Icon
Chapter 3 Vector Spaces 6 Section 3.4 Bases Objective •Whatisa basis for a vector space? • How to show that a set is a basis? • How to find a basis for a vector space? •Whatare coordinate vectors ?
Background image of page 6
Chapter 3 Vector spaces 7 Color mixing Three primary colors: Red , Green , Blue (RGB) Different color shade combination gives “all” colors -span{ Red , Green , Blue } = Color space -{ Red , Green , Blue } is linearly independent e.g. 20% Red + 45% Green + 30% Blue The three primary colors span the color space : None of the three primary colors are redundant : An Analogy
Background image of page 7

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

View Full DocumentRight Arrow Icon
Chapter 3 Vector spaces 8 Example Standard basis vectors for R 3 S is a smallest possible subset of R 3 so that every vector in R 3 is a linear combination of the elements in S . e 1 = (1, 0, 0), e 2 = (0, 1, 0), e 3 = (0, 0, 1) S = { e 1 , e 2 , e 3 } is called a basis for R 3 -span{ e 1 , e 2 , e 3 } = R 3 -{ e 1 , e 2 , e 3 } is linearly independent building block What is a basis? e.g. (2, 3, -5) = 2 e 1 + 3 e 2 -5 e 3 No redundant vectors
Background image of page 8
Chapter 3 Vector spaces 9 Definition 3.4.2 Let S = { u 1 , u 2 , …, u k } be a subset of R n .
Background image of page 9

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

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

Page1 / 35

lecture12 - Rn Fill in the blanks u1, u2, , uk c1u1+ c2u2 +...

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

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