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

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.

Sources

• Seth Warner: Modern Algebra (1965): $\S 27$