# Basis for a Vector Space

## Basis for a Vector Space

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: $$\overrightarrow{w} \space = \space a \overrightarrow{u} \space + \space b \overrightarrow{v}$$.