12240/Classnotes for Thursday September 27

Vector Spaces
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
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 C V, W != {emptyset}, is a subspace iff it is closed under the operations of V.
1. forall x, y elementof W, x + y elementof W
2. forall c elementof F, forall x elementof W, cx elementof W