Difference between revisions of "12240/Classnotes for Thursday September 27"
From Drorbn
Line 13:  Line 13:  
VS3. 0 vector  VS3. 0 vector  
−  VS4. + inverse  +  VS4. + inverse →  
VS5. 1x = x  VS5. 1x = x  
Line 25:  Line 25:  
== Theorems ==  == Theorems ==  
−  1.a x + z = y + z  +  1.a x + z = y + z ⇒ x = y 
−  1.b ax = ay, a  +  1.b ax = ay, a ≠ 0, ⇒ x = y 
−  1.c ax = bx, x  +  1.c ax = bx, x ≠ 0, ⇒ a = b 
Line 39:  Line 39:  
−  4. 0_F  +  4. 0_F ∙ x = 0_V 
−  5. a  +  5. a ∙ 0_V = 0_V 
Line 48:  Line 48:  
−  7. cx = 0  +  7. cx = 0 ⇔ c = 0 or x = 0_V 
Line 62:  Line 62:  
−  Theorem: A subset W  +  Theorem: A subset W ⊂ V, W ≠ {∅}, is a subspace iff it is closed under the operations of V. 
−  1.  +  1. ∀ x, y ∈ W, x + y ∈ W 
−  2.  +  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