lecture11 - Lecture Lecture 11 3.3 Linear Independence...

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

View Full Document Right Arrow Icon
ecture 11 Lecture 11 3.3 Linear Independence Chapter 3 Vector Spaces 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
Announcement ± This week : Lab + Tutorial + Practice session ± Next Week : Mid-term Test !!! ² Date: March 3 (Wednesday, seated by 6pm) ² Venue: MPSH1 (seat plan) ² Scope: Up to section 3.2 (Tutorials 1-4) ² Format: 4 questions do all ² Calculator: Scientific Calculator (NUS/SM2) raphing Calculator 3 students only) Graphing Calculator (H3 students only) ² Helpsheet: Self-prepared (NUS/ SM2) Will be provided (H3 students only) Lecture 11 Announcement 2 ² Bring your matric/student cards
Background image of page 2
From previous lecture Linear Span & Subspace A linear span of a set of vectors { v , v , …, v } in R n 1 2 k • it is a subset of R n denoted by span{ v 1 , v 2 , …, v k } • it contains all linear combinations of v 1 , v 2 , …, v k •it ±“ extends ” the vectors to a linear object (that contains the origin) A subset of R n is a subspace : ± is ±a ± linear span of some vectors in R n • it must contain the zero vector ± ± closed under addition Chapter 3 Vector Spaces 3 ± ± closed under scalar multiplication
Background image of page 3

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

View Full DocumentRight Arrow Icon
From previous lecture Subspaces of R 2 y y y x x x y y y x x x Chapter 3 Vector Spaces 4
Background image of page 4
ection 3 3 Section 3.3 Linear Independence Objective hat is a nearly independent/dependent et? • What is a linearly independent/dependent set? • How to show that a set is linearly (in)dependent? • What are some conditions on linearly (in)dependent sets? Other terminologies Chapter 3 Vector Spaces 5 “redundant” vectors
Background image of page 5

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

View Full DocumentRight Arrow Icon
What is a redundant vector in span( S )? Discussion 3.3.1 S = { (1, 1, 1) } S = { (1, 1, 1), ( 1, 1, 1) } 1 { ( ,,) 2 span( S 2 ) span( S 1 ) all scalar multiples of (1, 1, 1) all linear combination a (1, 1, 1) + b (-1, -1, -1) Adding the vector (–1, –1, –1) to S 1 does not change the linear span of S 1 There is a “ redundant ” vector in the span of S 2 Chapter 3 Vector spaces 6 (1, 1, 1) (-1, -1, -1)
Background image of page 6
What is a redundant vector in span( S )? Example S = { (1,1,1), (1,0,-2) } S = { (1,1,1), (1,0, 2), (2,3,5) } 1 { (,,) ,(,, ) 2 span( S 2 ) span( S 1 ) all linear combination a (1,1,1) + b (1,0,-2) all linear combination a (1,1,1) + b (1,0,-2) + c (2,3,5) Adding the vector (2, 3, 5) to S 1 does not change the linear span of S 1 3 (1,1,1) +( -1 ) (1,0,-2) There is a “ redundant ” vector in the span of S 2 (1,0,-2) Chapter 3 Vector spaces 7 (1,1,1) (2,3,5)
Background image of page 7

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

View Full DocumentRight Arrow Icon
What is a redundant vector in span( S )? Problem 3.2.13 Suppose u 1 , u 2 , …, u k are vectors taken from R n . If u k is a linear combination of u 1 , u 2 , …, u k –1 , then u k = d 1 u 1 + d 2 u 2 + ··· + d k-1 u k-1 { u 1 , u 2 , …, u k –1 } { u 1 , u 2 , …, u k –1 , u k } span span = c 1 u 1 + c 2 u 2 + ··· + c k-1 u k-1 c 1 u 1 + c 2 u 2 + ··· + c k-1 u k-1 + c k u k We say u k is a “ redundant ” vector in span{ u 1 , u 2 , …, u k –1 , u k }. Chapter 3 Vector spaces 8 If u œ span(S) , then span(S) = span(S » u )
Background image of page 8
Homogeneous system in vector equation form Vector equation (Example 1) , 0, 2 = 2 1, 1,0 = v ,1, 4 ( ) 1 1, 0, 2 v ( ) ,, ( ) 3 1,1, 4 v ariable scalars n 0 v v v = + + 3 3 2 2 1 1 c c c vector equation ( ) ( ) ( ) ( ) 12 3 1,0,2 1,1,4 0,0,0 cc c +− + = 0 0 0 2 -1 -1 3 2 1 c , c , c variable scalars (in R ) Can we find coefficients c , c , c that satisfies this homogeneous system in variables c 1 , c 2 , c 3 1 , 2 , 3 vector equation?
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.

Page1 / 34

lecture11 - Lecture Lecture 11 3.3 Linear Independence...

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