Sufficient Conditions for Basis of Finite Dimensional Vector Space/Examples

Examples of Use of Sufficient Conditions for Basis of Finite Dimensional Vector Space

Example: $2$ Dimensions

Let $V$ be a vector space of $2$ dimensions.

Let $\mathbf u, \mathbf v \in V$ be vectors

Let $\mathbf u$ and $\mathbf v$ be such that neither $\mathbf u$ nor $\mathbf v$ is a scalar multiple of the other..

Let $\mathbf w \in V$.

Then there exist scalars such that:

$\mathbf w = a \mathbf u + b \mathbf v$