Ordered Basis for Coordinate Plane

Theorem
Let $$a_1, a_2 \in \R^2$$ such that $$\left\{{a_1, a_2}\right\}$$ forms a linearly independent set.

Then $$\left({a_1, a_2}\right)$$ is an ordered basis for the $\R$-vector space $$\R^2$$.

Hence the points on the plane can be uniquely identified by means of linear combinations of $$a_1$$ and $$a_2$$.

Proof

 * OrderedBasisForPlane.png

Let $$P$$ be any point in the plane for which we want to provide a linear combination of $$a_1$$ and $$a_2$$.

Let the distance from $$O$$ to the point determined by $$a_1$$ be defined as being $$1$$ unit of length on the line $$L_1$$.

Let the distance from $$O$$ to the point determined by $$a_2$$ be defined as being $$1$$ unit of length on the line $$L_2$$.

Draw lines parallel to $$L_1$$ and $$L_2$$ through $$P$$

Then the coordinates $$\lambda_1$$ and $$\lambda_2$$ of $$P$$ are given by:
 * $$P = \lambda_1 a_1 + \lambda_2 a_2$$

by the Parallelogram Law.