12-240/Classnotes for Thursday September 27

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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