V,
(r,v) 
rv,
satisfying
A vector space over a field F is an
additive group (V,+,0)
with a scalar multiplication
F × V
V,
(r,v) rv,
satisfying
A linear combination of
a set of vectors
v1, ...,
vm in V is an expression of the form
i=1,...m
ri
vi where
r1, ...,
rm are elements of F.
A subspace of
V is a nonempty subset W of V that is closed under
all operations (so rv is in it if r in F and
v in W).
The linear combinations of a set of vectors
X in V
form a linear subspace, called the linear subspace
of V spanned by X.
A minimal set of generators of V is called a basis.
Example: The standard basis eiT (i= 1, ..., n) of Fn with en = (0, ...,0,1,0, ...,0) (the 1 appears in the i-th coordinate).
The number of elements of a basis is called the
dimension, notation dim(V).
Each vector space over F of dimension n is isomorphic to
Fn.
A linear map is a map f : V
W with f(rx+sy) =
r f(x) + sf(y) for all x
and y in V. If x1, ...,
xn is a basis of V, and
y1, ..., ym is a basis
of W, then the matrix of the linear map
f is the array of coefficients fi,j
such that f(xj) = ifi,jyi.
So, to each linear map we can assign a matrix once we have a bases. Conversely,
for fixed bases of V en W, each matrix ( of the right
dimensions) determines a linear map.
Take
V = Fn
and W = Fm. Then the matrix M =
(mi,j)i,j
can be viewed as the map that sends the column vector
v = (v1, ..., vn)T
to Mv, the matrix product of
Mv and v. The implicit choice of bases is the
standard bases.
By Hom(U,V) we denote
the vector space of all linear maps U V.
("Hom" is short for Homomorphisms.)
In plaats of Hom(U,U) schrijven we
ook End(U).
("End" is short for Endomorphisms.)
The latter is not just a vector space: it is a ring whose
multiplication is the composition of maps.
An element f of Hom(U,V) has a kernel Ker(f) = { x in V | f(x) = 0} and an image Im(f) = { f(x) | x in V }. The kernel is a subspace of V, the image is a linear subspace of W. The dimension of the image is called the rank of f. The map is injective if and only if the kernel is trivial (i.e., the linear subspace {0}). In general, dim(Ker(f)) + dim(Im(f)) = dim(V).
If a linear map in Hom(V,W) is bijective, then the inverse is also a linear map, and V and W are called isomorphic. They only differ in name, their structure as vector spaces is identical.
If dim(W) is finite, then a linear map in Hom(V,W) is bijective if and only if the kernel is trivial en and the rank is equal to dim(W).
An element f of End(V) is invertible if and only if its determinant, notation det(f), is nonzero. This is also equivalent to Ker(f) = {0}. By GL(U) we denote the group of all invertible elements in the ring End(U). ("GL" abbreviates "General Linear".)
A vector space U is called a sum
of two linear subspaces V and W if
U = V + W (in other words, V
is spanned by V 
W)
and it is called a direct sum if, in addition,
V 
W = 0; notation
U = V 
W.
If W is a linear subspace of V, then
V/W = { v+W | v in V} is a vector space, the
quotient space of V by W. It is the image
of the surjective linear map V V/W, v v+W with kernel W.
The dual vector space of a vector space V over F
is
Hom(V,F). Often we write
just V*.
If n = dim(V) is finite, then dim(V*) = n, so V is isomorphic to its dual.
The basis vj* (j = 1, ..., n) of V* is called the dual basis of v1, ..., vn if vj*(vk) = 1 if j = k and 0 otherwise.
If dim(V) is infinite, then V* is not isomorphic to V.