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

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{{12-240/Navigation}}
 
== Reminders ==
 
== Reminders ==
 
Web Fact: No link, doesn't exist!
 
Web Fact: No link, doesn't exist!
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Life Fact: Dror doesn't do email math!
 
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.
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Riddle: Professor and lion in a ring with <math>V_p = V_l</math>, help the professor live as long as possible.
  
 
== Recap ==  
 
== Recap ==  
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Base - what were doing today
 
Base - what were doing today
  
Linear combination (lc) - We say v is a linear combination of a set S = {u1 ... un}  if v = a1u1 ... anun for scalars from a field F.
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Linear combination (lc) - We say <math>v</math> is a linear combination of a set <math>S = \{u_1, \dots, u_n\}</math> if <math>v = a_1u_1, \dots, a_nu_n</math> for scalars from a field <math>F</math>.
  
Span - span(S) is the set of all linear combination of set S
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Span - <math>\operatorname{span}(S)</math> is the set of all linear combinations of the set <math>S</math>.
  
Generate - We say S generates a vector space V is span(S) = V
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Generate - We say <math>S</math> generates a vector space <math>V</math> is <math>\operatorname{span}(S) = V</math>.
  
 
== Pre - Basis ==
 
== Pre - Basis ==
  
'''Linear dependance'''
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'''Linear dependence'''
  
'''Definition''' A set S ⊂ V is called linearly dependant 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.
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'''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 independant.'''
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Otherwise, we call S '''linearly independent.'''
  
 
=== Examples ===
 
=== Examples ===
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u1 - 2u2 + u3 = 0
 
u1 - 2u2 + u3 = 0
  
S is linearly dependant.
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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.
 
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.
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Claim: This set is linearly independent.
 
Claim: This set is linearly independent.
  
Proof: Suppose (∑ ai*ei) = 0 ({ei} is linearly dependant.)
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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
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(∑ ai∙ei) = 0 ⇔ a1(1, 0, ..., 0) + ... + an(0, ..., 0, 1) = 0 ⇔ (a1, 0, ... , 0) + ... +  (0, ... , an) = 0
  
 
⇒ a1 = a2 = ... = an = 0!
 
⇒ a1 = a2 = ... = an = 0!
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===Comments ===
 
===Comments ===
  
1. {u} is linearly independant.
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1. {u} is linearly independent.
 
Proof:
 
Proof:
 
⇐ If u≠0, suppose au =0  
 
⇐ If u≠0, suppose au =0  
By property (a*u = 0, a = 0 or u = 0), a = 0. Thus, {u} is linearly independent.
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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.
 
⇒ By definition, au = 0 for {u} only when a = 0.
  
2. ∅ is linearly independant.
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2. ∅ is linearly independent.
  
  
 
Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.
 
Exercise: Prove: '''Theorem''' Suppose S1 ⊂ S2 ⊂ V.
  
-> If S1 is linearly dependant, then S2 is dependant.
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-> If S1 is linearly dependent, then S2 is dependent.
  
 
-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)
 
-> If S2 is linearly independent, then S1 is linearly independent. (Hint: contrapositive)
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== Basis ==  
 
== Basis ==  
  
'''Definition''': A subset  β is called a basis if 1. β generates V → span(β) = V and 2. β is linearly independent.
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'''Definition''': A subset  β is called a basis if (1) β generates V → span(β) = V and (2) β is linearly independent.
  
  
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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)
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(a,b) = (a+b)/2 (1, 1) + (a-b)/2 (1, -1)
 +
 
 +
== Lecture notes scanned by [[User:starash|starash]] ==
 +
<gallery>
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Image:12-240-1004-1.jpg|Page 1
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Image:12-240-1004-2.jpg|Page 2
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</gallery></math>

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

</math>