Let $k$ be a field, let $V$ be a [finite-dimensional vector space](/page/Finite-Dimensional%20Vector%20Space) over $k$, and let $V^*:=\operatorname{Hom}_k(V,k)$ be the dual [vector space](/page/Vector%20Space) of $V$. Then $V^*$ is finite-dimensional and $\dim_k V^*=\dim_k V$.
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More precisely, if $\mathcal{B}=(v_1,\ldots,v_n)$ is an ordered basis of $V$ and $\mathcal{B}^*=(v_1^*,\ldots,v_n^*)$ is the dual family defined by $v_i^*(v_j)=\delta_{ij}$ for all $i,j\in\{1,\ldots,n\}$, then $\mathcal{B}^*$ is an ordered basis of $V^*$.