General Vector Spaces

In general vector spaces, the idea of associating a column matrix with a vector leads to the concept of vector spaces. Let E^n denote the set of all n xx 1 column matrices, or vectors with n elements. The nonempty set E^n is a vector space because it satisfies the following axioms for vectors u, v, w in E^n and scalars a, b.

Basic Rule for General Vector Spaces:

Closure

u + v in E^n, au in E^n

Addition

Associative law

(u + v) + w = u + (v + w)

Zero vector

u + 0 = u

Existence of negatives

u + (-u) = 0

Commutative law

u + v = v + u

Multiplication

Distributive law

a(u + v) = au + av

(a + b)u = au + bu

Associative law

(ab)u = a(bu)

Unity law

1u = u
More about Vector Space for General Vector Spaces:

In general vector spaces, a set of vectors {u^(1), u^(2), . . . , u^(n)} is linearly dependent if and only if the sum

sum_(i=1)^n c_iu(i) = 0

is satisfied for a set of scalars {c_1, . . . ,c_m} that are not all zero. The set of vectors {u^(1), u^(2), . . . ,u^(n)} is linearly independent if above equation is satisfied only when c_i = 0 for all values of i.

A basis for the set E^n is a set of n linearly independent vectors that belong to E^n. If we write the basis as {u^(1), u(2), . . . , u(n)}, then any vector u in E^n can be written as a linear combination, or superposition, of basis vectors such that

u = sum_(i=1)^nc_iu^((i))

where {c_i} is a unique set of numbers for general vector spaces.

The inner product of two n-dimensional vectors u, v is defined as

(u, v) = sum_(i=1)^n u_i * v_i = (u*)^T = u xx v

where * denotes complex conjugation. Two vectors u^(i), u^(j) are orthogonal if their inner product is zero; that is, (u^(i), u^(j))=0. Vectors u^(i), u^(j) are orthonormal if their inner product satisfies

(u^(i), u^(J)) = delta_(ij)

Where the Kronecker delta is defined by

if i!= j then delta_ij = 0.

if i= j then delta_ij = 1.

The Euclidean length of a vector u with elements {u_i: i = 1, . . . , n} is the inner product of vector u with itself; thus

||u|| = (u, u)^(1/2) = [sum_(i=1)^(n) (| u_i |)^(2)]^(1/2)

The Euclidean lengths of two vectors u, v satisfy the Schwartz inequality

|u xxv| <= ||u|| ||v|| for general vector spaces.