**Note:** An integer linear combination is a linear combination in which all the weights (c’s) are integers.

**Span**

The span of a given list of vectors is defined as the set of all vectors which can be written as the linear combination of the given list of vectors.

Ex: \(\begin{bmatrix}x\\y\end{bmatrix}\) is said to be in the span of the vector-list \(\left\{\begin{bmatrix}x_1\\y_1\end{bmatrix},\begin{bmatrix}x_2\\y_2\end{bmatrix}, \begin{bmatrix}x_3\\y_3\end{bmatrix}\right\}\) if:

\(\begin{bmatrix}x\\y\end{bmatrix} = \alpha \begin{bmatrix}x_1\\y_1\end{bmatrix}+\beta \begin{bmatrix}x_2\\y_2\end{bmatrix}+\gamma \begin{bmatrix}x_3\\y_3\end{bmatrix}\)

where \(\alpha, \beta, \gamma\) ∈ \(\mathbb{R}\).

Let us solve a problem to improve our understanding.

**Q: Determine whether the vector \(\begin{bmatrix}19\\10\\-1\end{bmatrix}\) lies in the span of set of vectors: \(S = \left\{\begin{bmatrix}3\\-1\\2\end{bmatrix}, \begin{bmatrix}-5\\0\\1\end{bmatrix}, \begin{bmatrix}1\\7\\-4\end{bmatrix}\right\}\).**

**Approach:**

Let us first assume the equation: \(c_1v_1+c_2v_2+c_3v_3 = v\), where \(c_1, c_2, c_3\) are real numbers and try to find the solution \([c_1, c_2, c_3]\) if exists.