# 12-240/Classnotes for Thursday September 27

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

## 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