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14.1 Vector Spaces Definition 14.2 (See Definition on p. 190). A vector space is a non- empty set V of objects, called vectors, equipped with an addition operation “+” Let us recall the following basic properties about Rn. and scalar (=R or maybe C) multiplication “·” satisfying all of the properties above: i.e. For all u, v, w ∈ V and a, b ∈ R; Theorem 14.1 (See middle of p. 27). For all x, y, z ∈ Rn and a, b ∈ R; 1. Associativity of addition: u + (v + w) = (u + v) + w. 1. Associativity of addition: x + (y + z) = (x + y) + z. 2. Commutativity of addition: v + w = w + v. 2. Commutativity of addition: y + z = z + y. 3. Identity element of addition: There is a element 0 ∈ V such that 3. Identity element of addition: 0 + y = y for all y. 4. Inverse elements of addition: −y + y = 0 for all y ∈ Rn. 0 + v = v for all v ∈ V. 5. Distributivity of scalar multiplication with respect to vector addi- tion: a(y + z) = ay + az. 4. Inverse elements of addition: −v + v = 0 for all v ∈ V. 6. Distributivity of scalar multiplication with respect to field addition (In fact −v = (−1) · v.) (a + b)y = ay + by. 5. Distributivity of scalar multiplication with vector addition: 7. Compatibility of scalar multiplication with the multiplication on R a(by) = (ab)y. a · (v + w) = a · v + a · w. 8. Identity element of scalar multiplication 1y = y for all y ∈ Rn. 6. Distributivity of scalar multiplication with respect to field addi- We now turn these properties into a definition. tion (a + b) · v = a · v + b · v. 7. Compatibility of scalar multiplication with the multiplication on R a · (b · v) = (ab) · v. 8. Identity element of scalar multiplication 1 · v = v for all v ∈ V.

Example 14.3. The infinite blackboard with geometric addition and scalar mul- tiplication. a + b (a + b) + c = a + (b + c)

a

b

b + c

c Fig. 14.1. Checking the graphical vector addition is associative.

Example 14.4 (The Main Umbrella Example). Let D be a non-empty set and let V = V (D) denote all functions, f : D → R. For f, g ∈ V and λ ∈ R we define f + g and λ · f by

(f + g)(t) = f (t) + g (t) (addition in R), (λ · f)(t) = λf (t) (multiplication in R). It can now be checked that V is a vector space so that functions have now become vectors! Essentially all other examples of vector spaces we give will be related to an example of this form.

Example 14.5. R3 = {x : {1, 2, 3} → R} and more generally Rn = {x : {1, 2, . . . , n} → R} . Example 14.6. The vector space of 2 × 2 matrices;

M2×2 = {A : A is a 2 × 2 – matrix } = {A : {(1, 1) , (1, 2) , (2, 1) , (2, 2)} → R} . This can be generalized. Definition 14.7 (Subspaces are like Las Vegas). Let V be a vector space. A non-empty subset, H ⊂ V, is a subspace of V if H is closed under addition and scalar multiplication. Note, if H is a subspace and v ∈ H, then 0 = 0·v ∈ H. [What happens in Vegas stays in Vegas and 0 ∈ Vegas.] Proposition 14.8. Every subspace H of a vector space, V, is a vector space!