Definition:

If \( S = \{ \overrightarrow{v_1}, \dotsc \} \) is a non-empty set of vectors then the **span of S** is a set containing **all possible linear combinations** of \( S \).

To determine the span of a vector let us consider all of its possible linear combinations. A linear combination of a set of vectors involves scaling and them taking their sum. Because we only have one vector we can only scale it as there is no other vector to sum it with. This means that the set of all possible linear combinations of a vector is equivalent to its scalar multiples.

Because scaling a vector does not change its direction but simply stretches it the set containing all of its scalar multiples forms a line in the direction defined by the vector.

**The span of any non-zero vector is a line.**

The span of a set of containing more than one vector can be a line, plane or volume depending on how many of them are linearly dependent.

Definition:

If \( S = \{ \overrightarrow{v_1} , \overrightarrow{v_2}, \dotsc \} \) is a set of two or more vectors in a vector space V, then S is said to be a **linearly independent set** if no vector in S can be expressed as a **linear combination** of the others. A set that is not linearly independent is said to be **linearly dependent.**

**If a vector, v, is linearly dependent with respect to a set of vectors S then v can be expressed as a linear combination of the set S.**

If two vectors are linearly dependent then we know either can be expressed as a linear combination/scalar multiple of the other. Geometrically this can be viewed as both vectors lying on the same line. In a sense one of the vectors is redundant, it is already included in the span of the other and as such the span of two linearly dependent vectors is still a line.

If two vectors are linearly independent then neither can be expressed as a linear combination of the other. Geometrically this means that the two vectors do not lie on the same line, because of this both vectors contribute to the span of their set. The span of two linearly independent vectors is a plane.

Example:

An example of two linearly independent vectors are the two basis vectors \( \overrightarrow{i} \) and \( \overrightarrow{j} \). Because they are linearly independent any vector on the cartesian plane can be expressed as a linear combination of the two vectors. \( \overrightarrow{i} \) and \( \overrightarrow{j} \) span the cartesian plane.

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

The span of set of vectors can sometimes be a line, plane or volume.The span of any non-zero vector is a line, the span of two linearly independent vectors is a plane, and the span of three linearly independent vectors is a volume.

The span of a set of vectors is an infinite set containing all possible linear combinations. The visualization to the left shows a subset of the entire span by randomly generating 150 linear combinations of the following set of vectors. (Each line is a linear combination)

- a. 1 vector in \( \mathbb{R^2} \)
- span = line

- b. 2 linearly independent vectors in \( \mathbb{R^2} \)
- span = plane

- c. 2 linearly independent vectors in \( \mathbb{R^3} \)
- span = plane

- c. 3 linearly independent vectors in \( \mathbb{R^3} \)
- span = volume

**Visualize the span of up to three vectors with the interactive program below.**

Once you have edited the components of your vectors using their sliders press the button to **generate 250 linear combinations** and press the button to clear them.

To edit the number of vectors being visualized use the and buttons and change the vector space by using the select drop-down labeled **Change Dimension**.

\( \mathbb{R^1} \)

\( \mathbb{R^2} \)

\( \mathbb{R^3} \)

...

\( span(V) \)