The column space of an matrix is the span of
the columns of . It is a subspace
of and we denote it by
col().
Determine whether and are in the column space of We need to solve the two vector equations of the form where first being
, then . The respective reduced
row-echelon forms (简化行阶梯形式) of the augmented matrices
corresponding to the two systems are Therefore we can find scalars , and holds when , , but not when . From this we deduce that
is in col(), but is not.
Recall that the system of linear
equations in unknowns can be
written in linear combination form: Note that the left side of this equation is simply a linear
combination of the columns of ,
with the scalars being the components of . The system will have a solution
if, and only if, can be
written as a linear combination of the columns of . Stated another way, we have the
following:
Theorem: A system has a solution (meaning
at least one solution) if, and only if, is in the column space of .
Let’s consider now only the case where , so we have linear
equations in unknowns. We have
the following facts:
If col() is all of , then will have a solution for
any vector . What’s more, the
solution will be unique.
If col() is a proper subspace
of (that is, it is not
all of R n), then the equation will have a solution if, and only if, is in col(). If is in col() the system will have infinitely many
solutions.
Next we define the null space of a matrix.
Definition: Null Space of a Matrix
The null space of an
matrix is the set of all
solutions to . It is a
subspace of and is
denoted by null().
Determine whether and are in the column space of A vector is in the
null space of a matrix if . We see that
so is in the null() and is not.
Still considering only the case where , we have the following fact about the null space:
If null() is just the zero
vector, is invertible and has a unique solution
for any vector .
