The Unique Representation Theorem

Let $\mathcal{B} = \{ \vec{b}_1, \dots, \vec{b}_n \}$ be a basis for a vector space $V$. Then for each $\vec{x} \in V$, there exists a unique set of scalars $c_1, \dots, c_n$ such that

$$\vec{x} = c_1 \vec{b}_1 + \cdots + c_n \vec{b}_n$$

Theorem

Let $\mathcal{B} = \{ \vec{b}_1, \dots, \vec{b}_n \}$ be a basis for a vector space $V$. Then the coordinate mapping $\vec{x} \mapsto [\vec{x}]_\mathcal{B}$ is a one-to-one linear transformation from $V$ onto $\mathbb{R}^n$.