LINEAR ALGEBRA (Math 211)
Sheet #6
VECTOR SPACES

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) 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