Difference between revisions of "12-240/Classnotes for Thursday October 4"

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(Basis)
(Interesting inequality)
 
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This holds is true if the field does not have  characteristic 2. Can you see why?
 
This holds is true if the field does not have  characteristic 2. Can you see why?
  
(a,b) = (a+b)/2 * (1, 1) + (a-b)/2 * (1, -1)
+
(a,b) = (a+b)/2 (1, 1) + (a-b)/2 (1, -1)
  
 
== Lecture notes scanned by [[User:starash|starash]] ==
 
== Lecture notes scanned by [[User:starash|starash]] ==

Latest revision as of 05:46, 7 December 2012

Contents

Reminders

Web Fact: No link, doesn't exist!

Life Fact: Dror doesn't do email math!

Riddle: Professor and lion in a ring with V_p = V_l, help the professor live as long as possible.

Recap

Base - what were doing today

Linear combination (lc) - We say v is a linear combination of a set S = \{u_1, \dots, u_n\} if v = a_1u_1, \dots, a_nu_n for scalars from a field F.

Span - \operatorname{span}(S) is the set of all linear combinations of the set S.

Generate - We say S generates a vector space V is \operatorname{span}(S) = V.

Pre - Basis

Linear dependence

Definition A set S ⊂ V is called linearly dependent if you can express the zero vector as a linear combination of distinct vectors from S, excluding the non-trivial linear combination where all of the scalars are 0.


Otherwise, we call S linearly independent.

Examples

1. In R^3, take S = {u1 = (1,4,7), u2 = (2,5,8), u3 = (3,6,9)}

u1 - 2u2 + u3 = 0

S is linearly dependent.

2. In R^n, take {e_i} = {0, 0, ... 1, 0, 0, 0} where 1 is in the ith position, and (ei) is a vector with n entries.

Claim: This set is linearly independent.

Proof: Suppose (∑ ai∙ei) = 0 ({ei} is linearly dependent.)

(∑ ai∙ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... + (0, ... , an) = 0

⇒ a1 = a2 = ... = an = 0!

Comments

1. {u} is linearly independent. Proof: ⇐ If u≠0, suppose au =0 By property (a∙u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.

⇒ By definition, au = 0 for {u} only when a = 0.

2. ∅ is linearly independent.


Exercise: Prove: Theorem Suppose S1 ⊂ S2 ⊂ V.

-> If S1 is linearly dependent, then S2 is dependent.

-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)

Basis

Definition: A subset β is called a basis if (1) β generates V → span(β) = V and (2) β is linearly independent.


Examples

1. V = {0}, β = {}

2. {ei} for F^n, this is what we call the standard basis

3. B = {(1,1),(1, -1)} is a basis for R^2

4. P_n(F) β = {x^n, x^n-1, ... , x^1, x^0}

5. P(F), β = (x^0, x^1 ... and on} (Infinite basis!)

Interesting inequality

This holds is true if the field does not have characteristic 2. Can you see why?

(a,b) = (a+b)/2 ∙ (1, 1) + (a-b)/2 ∙ (1, -1)

Lecture notes scanned by starash

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