MATH 240: Vector Spaces

Definition: A vector space is a set V on which two operations + and · are defined, called vector addition and scalar multiplication.

The operation + (vector addition) must satisfy the following conditions:

Closure: If u and v are any vectors in V, then the sum   u + v   belongs to V.

(1) Commutative law: For all vectors u and v in V,     u + v = v + u

(2) Associative law: For all vectors u, v, w in V,     u + (v + w) = (u + v) + w

(3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V,     0 + v = v   and   v + 0 = v.

(4) Additive inverses: For each vector v in V, the equations     v + x = 0   and   x + v = 0     have a solution x in V, called an additive inverse of v, and denoted by - v.
The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:

Closure: If v in any vector in V, and c is any real number, then the product   c · v   belongs to V.

(5) Distributive law: For all real numbers c and all vectors u, v in V,     c · (u + v) = c · u + c · v

(6) Distributive law: For all real numbers c, d and all vectors v in V,     (c+d) · v = c · v + d · v

(7) Associative law: For all real numbers c,d and all vectors v in V,     c · (d · v) = (cd) · v

(8) Unitary law: For all vectors v in V,     1 · v = v

Subspaces

Definition: Let V be a vector space, and let W be a subset of V. If W is a vector space with respect to the operations in V, then W is called a subspace of V.

Theorem: Let V be a vector space, with operations + and   ·, and let W be a subset of V. Then W is a subspace of V if and only if the following conditions hold.

Sub0 W is nonempty: The zero vector belongs to W.

Sub1 Closure under +: If u and v are any vectors in W, then   u + v   is in W.

Sub2 Closure under ·: If v is any vector in W, and c is any real number, then   c · v   is in W.