Geometry of linear equations: structure of image and null space
Definition 1. The subspace notion allows us to describe the algebraic structure of the set of solutions of The special case
is called a homogeneous equation. Obviously,
satisfies it but there may be other solutions. The equation
is called an inhomogeneous equation. We address the questions of existence and uniqueness of its solutions.
Structure of the image of 
Recall Basic observation 1: The image is the set of
for which the inhomogeneous equation has solutions.
Exercise 1. is a linear subspace in
Proof. This follows from the first characterization of the matrix image. Here is a direct proof. Suppose Then there exist
such that
By linearity this gives
Thus for
we have found
such that
which means
for any
and
is a subspace.
Structure of the null space of 
Definition 2. The set of solutions of the homogeneous equation
is called the null space of
It is denoted
Exercise 2. The null space of is a linear subspace of
Proof. Suppose so that
Then by linearity
so
for any
and
is a linear subspace.
Description of the set of solutions of 
Intuition. In straight lines and planes that don't contain the origin can be obtained by shifting straight lines and planes that do (geometry should dominate the algebra at this point, see Figure 1). This is generalized in the next definition.

Figure 1. Shifting a subspace gives a hyperplane
Definition 3. Let be any vector and let
be a subspace.
denotes a shift of
by
and it is obtained by adding to
all elements of
Some people call
a hyperplane.
Exercise 3. As we know, the equation has solutions if and only if
Let us fix
and let
be some solution of
. Then the set of all solutions of this equation is
(in detail: any other solution
of that equation can be obtained by adding an element
of the null space to
:
). This is written as
Proof. Let and let
be any solution of
Then
and
for some
We obtain
which proves the inclusion
Conversely, if
with
, then
and we obtain
Conclusion. If , then
is one-to-one and we have uniqueness of solutions of the inhomogeneous equation; otherwise,
can serve as a measure of non-uniqueness.
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