Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

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.