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The Unique Representation Theorem

Let B={b1,,bn} \mathcal{B} = \{ \vec{b}_1, \dots, \vec{b}_n \} be a basis for a vector space V V . Then for each xV \vec{x} \in V , there exists a unique set of scalars c1,,cn c_1, \dots, c_n such that

x=c1b1++cnbn \vec{x} = c_1 \vec{b}_1 + \cdots + c_n \vec{b}_n

Theorem

Let B={b1,,bn} \mathcal{B} = \{ \vec{b}_1, \dots, \vec{b}_n \} be a basis for a vector space V V . Then the coordinate mapping x[x]B \vec{x} \mapsto [\vec{x}]_\mathcal{B} is a one-to-one linear transformation from V V onto Rn \mathbb{R}^n .