Vector space
F is a field.
A set V that is an Abelian group under addition with the
property that for each a  F , v V , there is an element
av V such that the following conditions hold for all
a, b  F and all u, v V :
1. a(u  v)  au  av .
2. (a  b)v  av  bv .
3. a(bv)  (ab)v .
4. 1v  v .
Scalar
A member of a field over which a vector space is
constructed.
Vector
A member of a vector space.
Subspace
V is a vector space over a field F.
A subset U of V that is a vector space over F under the
operations of V.
Linearly dependent
S is a set of vectors; F is a field.
Having the property that there are vectors v1,..., vn from
S and scalars a1,..., an from F, not all zero, such that
a1v1  ...  anvn  0 .
Linearly independent
Not linearly dependent.
Basis
V is a vector space over a field F.
A subset B of V that is linearly independent over F and
for which every element of V is a linear combination of
elements of B.
Invariance of Basis Size
If {u1, u2 ,..., um} and {w1, w2 ,..., wn} are both bases of a
vector space V, then m  n .
Dimension
The number of elements of a basis of a vector space.
For the trivial vector space, the number 0.