Although there are many operations on columns of real numbers, the fundamental operations in linear algebra are the linear ones: addition of two columns, multiplication of the whole column by a constant, and compositions of those operations. In this section we will introduce some vocabulary to help us reason about linear relationships between vectors.
where are real numbers. The 's are called the weights of the linear combination.
Suppose that and . Draw the set of all points in for which the vector can be written as an integer linear combination of and .
Note: An integer linear combination is a linear combination where the weights are integers.
Solution. A bit of experimentation reveals that the integer linear combinations of these two vectors form a lattice as shown.
Is in the span of and ?
Find values and such that . We have
We visualize a set of vectors in by associating the vector with the point —in other words, we associate each vector with the location of its head when its tail is drawn at the origin. Apply geometric reasoning to solve the following exercises.
The span of two vectors in
The span of three vectors in
Solution. The span of a list containing only the zero vector is just the origin. The span of a list containing a single vector is a line through the origin, since points in the same direction as for any . The span of a list containing two non-parallel vectors and is all of , since the span consists of the union of all lines which run in the direction and pass through any point in the span of . Including more vectors can't increase the span further, so these are the only possibilities. So the correct answer is (e).
The same reasoning implies that the span of a list of vectors in must be either the origin, or a line or plane through the origin, or all of . So the correct answer choice is the fourth one.
Check out the 3Blue1Brown video segment below for some helpful visualizations for spans of vectors in three-dimensional space.