12-240/Classnotes for Thursday September 27
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Vector Spaces
Reminders
- Tag yourself in the photo!
- Read along textbook 1.1 to 1.4
- Riddle: Professor in ring with lion around the perimeter. Consider this: http://mathforum.org/library/drmath/view/63421.html
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
1.a Same as for fields
1.b. Use similar proof as for fields, but use VS6 NOT F2b. F2b guarantees existence, but VS6 allows algebraic manipulation.
1.c Discussed after proof of 7, harder than you think at first glance.
2. Same as F.
3. Same as F
4. 0x + 0x = (0+0)x [VS8] = 0x = 0x + 0 [VS3] = 0 + 0x [VS1] => 0x + 0x = 0 + 0x => [Cancellation property] 0x = 0
5. Same as 4 except using 0_V + 0_V = 0_V and using VS7
6. Skip
7. Prove both ways: Easy way is to the left, show left is 0 if either on right is 0. To the right, Suppose c not= 0, then show x must equal 0.
1.c
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
