Here's how you get this linear combination.
Draw two lines through the head of the purple vector parallel to the red
and blue vectors. These lines intersect the lines of the vectors at points
which determine scalar multiples of those vectors (the pale red and pale
blue vectors in the diagram). The purple vector is the sum of those pale
vectors (using the parallelogram rule for addition), i.e. it is the sum
of scalar multiples of the red and blue vectors.

Suppose you have two vectors in 2-space (the red and blue vectors in
the diagram). You can change the purple vector by dragging its head to see
it written as a linear combination of the red and blue vectors. You can make the purple vector be any vector in 2-space and this will work, so the span of the red and blue vectors is all of 2-space.

The line of reasoning above works for any pair
of vectors in 2-space, *provided those vectors are not
parallel*. If the red and blue vectors were parallel, then the only linear
combinations of them you could get would have to be parallel to both of them
- you couldn't get anything else. To summarize:

Two non-parallel vectors in 2-space span the whole of 2-space. Two parallel vectors in 2-space span only the line of vectors containing them.

Now let's look at the span of two vectors in 3-space (again red and blue).

If the two vectors are parallel, then any linear combination of them must be parallel to both, so as before, the span of two parallel vectors consists of the line through the origin which contains the two vectors.

If the vectors are not parallel, then there is a single plane through the origin containing them. Any linear combination of those vectors lies in that plane and any vector in that plane is in their span (using the same sort of argument we used for 2-space). So two non-parallel vectors in 3-space span the plane of vectors containing them.

Vector Spans | |||

Introduction | Definition of a span | The span of two vectors | The span of three or more vectors |