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
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. A straight line or a plane that does not pass through the origin is not a subspace, see Figure 1. The straight line does not pass through the origin. If we take vectors from it, their sum, found by the parallelogram rule, does not belong to .
Definition 1. The set is called a linear subspace of if for any it contains the whole plane drawn through By induction, any linear combination of any elements of a subspace belongs to it.
The definition does not exclude the extreme cases and . These two cases are called trivial.
Sometimes we are interested in how a subspace is generated.
Definition 2. Take any vectors and consider the set of all linear combinations
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 We also say that is spanned by
In this terminology, the first characterization of the matrix image says that it is spanned by that matrix' columns.