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
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
