Vector Spaces

Vector Space

Let V be a set on which two operations, vector addition and scalar multiplication, are defined. If the listed axioms are satisfied for every for every u, v, w ∈ V and every scalar c, d, then V is called a vector space.

Addition

(i) u + v ∈ V (closure under addition)

(ii) u + v = v+u (commutative property)

(iii) u + (v + w) = (u+v)+w (associative property)

(iv) There exists a zero vector 0 ∈ V such that u + 0 = u (additive identity)

(v) For every u ∈ V, there exists −u ∈ V such that u+(−u) = 0 (additive inverse)

Scalar Multiplication

(vi) cu ∈ V (closure under scalar multiplication)

(vii) c(u + v) = cu + cv (distributive property)

(viii) (c + d)u = cu + du (distributive property)

(ix) c(du) = (cd)u (associative property)

(x) 1u = u (scalar identity)

4 Components of Vector Space

a set V, a set of scalars, vector addition, scalar multiplication. when deciding whether a set is a vector space, we must know not only the set V, but also how addition and scalar multiplication are defined.

ex: set of all ordered n-tuples of real numbers, Rⁿ = { (v₁,v₂,...,v_n) | v_i ∈ R }

R satisfies all 10 axioms, so it is a vector space.

Matrix Vector Space

M_{m, n} = { all m×n matrices with real entries is a vector space }

(image is example of M_{2, 3})

Polynomial Vector Space

P_n = { all polynomials to the nth degree with real numbers is a vector space }

(image is example of P₂)

Continuous Function Vector Space

C: Stands for Continuous

(-inf, +inf): Means the functions must be continuous across the entire real number line R

C(-inf, +inf) = { all continuous functions from negative infinity to positive infinity with real numbers is a vector space }

Frequently Mentioned Vector Spaces

Properties of Vector Space

Let v be any element of a vector space V, and let c be any scalar. Then the following properties hold:

(i) 0v = 0

(ii) c0 = 0

(iii) If cv = 0, then c = 0 or v = 0

(iv) (−1)v = −v

Checklist for if V is a Vector Space