## 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.

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 subspaces 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

(2) with

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** and use the notation

In this terminology, the first characterization of the matrix image says that it is spanned by that matrix' columns.

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