Linear combination and span

Linear combination

A linear combination of the vectors x, y and z is any vector of the form a x + b y + c z where a, b and c are constants.

Example: (8,3,3) is a linear combination of (1,1,1) and (1,0,0) since it satisfies the following equation with constants 3 and 5.

$$3\begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix} + 5\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix} =\begin{bmatrix}8 \\ 3 \\ 3 \end{bmatrix}$$

Linear span

The linear span of the vectors x and y is the set of all possible linear combinations of them.

Example: The linear span of (1,1,1) and (1,0,0) is the set of all vectors of the form

$$s\begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix} + t\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}$$

where s and t range over all possible scalar values.

Link: http://faculty.bard.edu/belk/math213s14/LinearCombinationsAndSpanRevised.pdf