Let's look at what it means geometrically for
a set of two or more vectors to be linearly dependent or linearly independent.

Suppose we have a set of two non-zero vectors.
One is a linear combination of the other whenever it is a scalar multiple of
the other, i.e. whenever it is parallel to the other. Thus:

A set of two vectors is linearly dependent if one is parallel to the other,

and linearly independent if they are not parallel.

(This is true in either 2-space or 3-space.)

Now suppose we have three non-zero vectors, either
in 2-space or in 3-space. There are three possible cases, illustrated for 3-space
below. (The vectors we're discussing are in shades of blue.)

If any two of the vectors are parallel, then one
is a scalar multiple of the other. A scalar multiple is a linear combination, so the vectors are **linearly dependent**.
(Notice that all three vectors also lie in a plane.)

If no two of the vectors are parallel but all
three lie in a plane, then any two of those vectors span that plane. The third
vector is a linear combination of the first two, since it also lies in this
plane, so the vectors are** linearly dependent**.

If the three vectors don't all lie in some plane
through the origin, none is in the span of the other two, so none is a linear
combination of the other two. The three vectors are **linearly
independent**.

Now suppose we have four or more non-zero vectors.
If any three all lie on a line or on a plane, they are clearly linearly dependent,
so the set of all the vectors is linearly dependent. If some three of the
four don't lie on a plane, they span all of 3-space, so the others must be
linear combinations of those three and again, the vectors are linearly dependent.
Thus:

Four or more vectors in 2-space or in 3-space must be linearly dependent.

Here's a summary of how geometric vectors in 2-space or in 3-space can
be linearly independent.

- Two vectors are linearly independent if they are not parallel.
- Three vectors are linearly independent if they don't all lie in a plane.
- More than two vectors in 2-space must be linearly dependent
- More than three vectors in 3-space must be linearly dependent.

Linear Independence | |||

Introduction | The basic definition of linear independence | Examples of linearly independent geometric vectors | A more practical form of linear independence |