The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).

Created almost 9 years ago, updated about 8 hours ago

The coordinate mapping onto $\mathbb F^n$ is surjective: for any tuple $(c_1,\ldots,c_n)^T$, the vector $v=\sum_ic_ib_i$ has exactly those coordinates. Together with injectivity and linearity, this proves that finite-dimensional spaces are coordinate-isomorphic to $\mathbb F^n$.

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