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 [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex] with different directions, you can obtain all the vectors in the plane. Therefore, [latex](\overrightarrow{u},\space \overrightarrow{v})[/latex] 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 [latex]\overrightarrow{u}[/latex], [latex]\overrightarrow{v}[/latex] and [latex]\overrightarrow{w}[/latex].
The sides of the parallelogram ABCD are scalar multiples of [latex]\overrightarrow{u}[/latex] and [latex]\overrightarrow{v}[/latex]. The construction below shows that: [latex]\overrightarrow{w} \space = \space a \overrightarrow{u} \space + \space b \overrightarrow{v}[/latex].
