Axiomatic definition. A (real) vector space V is a set
with two operations called addition:
that satisfy the
following axioms for all ,
from V and all
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)||RA||the subspace of spanned by the rows of A||(the row space of A)|
|(2)||CA||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