# The Dual Base for E* of a Finite-Dimensional Vector Space E

Let $E$ be a finite-dimensional vector space, say $\mathrm{dim} (E) = n$ and let $\{ e_1, e_2, ..., e_n \}$ be a basis for $E$. Then every $x \in E$ can be written uniquely in the form:

(1)For each $1 \leq i \leq n$, let $e_i^*$ be the linear form on $E$ defined for all $x \in E$ by:

(2)Then $\{ e_1^*, e_2^*, ..., e_n^* \}$ is a base for $E^*$. Indeed, $\{ e_1^*, e_2^*, ..., e_n^* \}$ is linearly independent, for if:

(3)for some $\mu_1, \mu_2, ..., \mu_n \in \mathbf{F}$, then for each $1 \leq j \leq n$:

(4)so that $\mu_1 = \mu_2 = ... = \mu_n = 0$.

Furthermore, $\{ e_1^*, e_2^*, ..., e_n^* \}$ spans $E^*$, since for each $f \in E^*$ we have that:

(5)So for each $1 \leq i \leq n$, let $\mu_i := \langle e_i, f \rangle$. Then from above we have that:

(6)Thus, $f = \sum_{i=1}^{n} \mu_i e_i^*$.