Geometry of linear equations: linear spaces and subspaces
is a linear space in the following sense: for any
and any
the linear combination
belongs to
From earlier exercises we know that
and
are straight lines drawn through the vectors
Therefore
is a plane drawn through those vectors. We can say that
is a linear space in the following sense: for any
, the whole plane drawn through
is contained in
Subsets of may have this property.
Example 1. On the plane, take any straight line passing through the origin (the slope doesn't matter). The equation of such a line is
(1)
where at least one of the coefficients is different from zero. If
we can solve the equation for
and get a more familiar form
If
we get a vertical line
If two points
satisfy (1), then you can check that their linear combination
also satisfies (1). A straight line that does not pass through the origin is described by
with
Example 2. In , take any straight line or any 2D plane passing through the origin. A straight line is described by a system of two equations
and a plane is described by one equation
You can do the algebra as above to show that linear combinations of elements of these sets belong to these sets. However, I suggest using the geometric interpretation of linear operations to show that these are examples of subspaces.

Figure 1. A hyperplane is not a subspace
A straight line or a plane that does not pass through the origin is not a subspace, see Figure 1. The straight line
Definition 1. The set
The definition does not exclude the extreme cases
Sometimes we are interested in how a subspace is generated.
Definition 2. Take any vectors
(2)
This set is a linear subspace (because a linear combination of expressions of type (2) is again of this type) and it is called a span of
In this terminology, the first characterization of the matrix image says that it is spanned by that matrix' columns.
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