Axiomatic definition. A (real) vector space V is a set
with two operations called addition:
and
scalar multiplication:
that satisfy the
following axioms for all ,
,
from V and all
scalars ,
from .
Definition. A subset W of a vector space V is called a subspace if W is itself a vector space under the addition and scalar multiplication defined on V
Example. Let , , ..., be vectors in in a vector space V. Then the set of all possible linear combinations of these vectors is a subspaces in V which is called the space spanned by , , ..., : .
Basis. Dimension is the number of vectors in a basis.
Vector spaces associated with a matrix.
For a matrix
with m rows and n columns one can
define the following four vector spaces
(1) | R_{A} | the subspace of spanned by the rows of A | (the row space of A) | |
(2) | C_{A} | the subspace of spanned by the columns of A | (the column space of A) | |
(3) | (the kernel (nullspace) of A) | |||
(4) | for some | (the image (range) of A) |
Relations: ; ; .
Definition. Rank of a matrix A is the dimension .
Theorem. Let
be a particular solution of a system of
linear equations
and let
,
,
...,
be a basis of the kernel of A. Then the general
solution of the system has the form