In a geometric space E, a linearly independent set of vectors such that every vector in E can be written as a linear combination of the vectors.
Through the linear combination of two vectors \(\overrightarrow{u}\) and \(\overrightarrow{v}\) with different directions, you can obtain all the vectors in the plane. Therefore, \((\overrightarrow{u},\space \overrightarrow{v})\) is said to be a basis for the set of vectors.
In other words, a basis for a vector space is a set of vectors that generates a vector space.
Example
Consider the vectors \(\overrightarrow{u}\), \(\overrightarrow{v}\) and \(\overrightarrow{w}\).
The sides of the parallelogram ABCD are scalar multiples of \(\overrightarrow{u}\) and \(\overrightarrow{v}\). The construction below shows that: \(\overrightarrow{w} \space = \space a \overrightarrow{u} \space + \space b \overrightarrow{v}\).
