Difference between revisions of "12-240/Classnotes for Thursday September 27"

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VS3.  0 vector
 
VS3.  0 vector
  
VS4. + inverse -> -
+
VS4. + inverse -
  
 
VS5. 1x = x
 
VS5. 1x = x
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== Theorems ==
 
== Theorems ==
  
1.a x + z = y + z => x = y
+
1.a x + z = y + z x = y
  
1.b ax = ay, a != 0, => x = y
+
1.b ax = ay, a 0, x = y
  
1.c ax = bx, x != 0, => a = b
+
1.c ax = bx, x 0, a = b
  
  
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4. 0_F * x = 0_V
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4. 0_F x = 0_V
  
  
5. a * 0_V = 0_V
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5. a 0_V = 0_V
  
  
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7. cx = 0 <=> c = 0 or x = 0_V
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7. cx = 0 c = 0 or x = 0_V
  
  
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Theorem: A subset W C V, W != {emptyset},  is a subspace iff it is closed under the operations of V.
+
Theorem: A subset W V, W {},  is a subspace iff it is closed under the operations of V.
  
1. forall x, y elementof W, x + y elementof W
+
1. x, y W, x + y W
  
2. forall c elementof F, forall x elementof W, cx elementof W
+
2. c F, x W, cx W

Revision as of 20:39, 27 September 2012

Vector Spaces

Contents

Vector space axioms

(Quick recap)

VS1. x + y = y + x

VS2. (x + y) + z = x + (y + z)

VS3. 0 vector

VS4. + inverse → -

VS5. 1x = x

VS6. a(bx) = (ab)x

VS7. a(x + y) = ax + ay

VS8. (a+b)x = ax + bx

Theorems

1.a x + z = y + z ⇒ x = y

1.b ax = ay, a ≠ 0, ⇒ x = y

1.c ax = bx, x ≠ 0, ⇒ a = b


2. 0 is unique.


3. Additive inverse is unique.


4. 0_F ∙ x = 0_V


5. a ∙ 0_V = 0_V


6. (-a) x = -(ax) = a(-x)


7. cx = 0 ⇔ c = 0 or x = 0_V


Rough sketches for proofs

Subspaces

Definition: Let V be a vector space over a field F. A subspace W of V is a subset of V, has the operations inherited from V and 0_V of V, is itself a vector space.


Theorem: A subset W ⊂ V, W ≠ {∅}, is a subspace iff it is closed under the operations of V.

1. ∀ x, y ∈ W, x + y ∈ W

2. ∀ c ∈ F, ∀ x ∈ W, cx ∈ W