) is a column of n real numbers.
Vectors can be multiplied by scalars, and added together. The algebraic
properties for
vectors are similar to the properties satisfied by scalars. A linear
combination of vectors is a sum of scalar multiples of the vectors. A
linear system is just a vector equation where the left-hand side of the
equation is a linear combination of some vectors with the unknowns for the
system used as scalar multipliers. The right-hand side of the equation is a
vector of right-hand sides for the linear system. The set of all possible
linear combinations of a set of vectors is called the span of the set. In two
(three) dimensions the span can interpreted geometrically as a line (plane).
In general, the span of a set of n-vectors is a subset of ; sometimes
the span includes all of and sometimes it does not.
The result of
multiplying
an n-vector 
by an 
matrix A is
an m-vector 
(
) that is a linear combination (with the 
's
as multipliers) of the columns of A.
The matrix equation 
is another way of writing down a linear
system whose coefficient matrix is A and right-hand side is 
. When
we try to solve a linear system, we are trying to find scalars so that
if we use these scalars as multipliers for the columns of A and add the
results together, then we get . Another way to consider this process is to
say that we are trying to determine if is in the span of the columns of
A. The columns of A span 
if and only if has a solution
for all possible 's in .