Let $\mathbb{F}$ be a field and let $V$ be a finite-dimensional $\mathbb{F}$-vector space. Then $\dim V^* = \dim V$.
AlgebraLinear Algebra
Discussion
The dual space of a finite-dimensional vector space has the same dimension as the original space. This is an immediate corollary of the Dual Basis theorem.
Proof
**Proof.** By the [Dual Basis](/theorems/414) theorem, if $V$ has a basis of $n$ vectors, then $V^*$ has a dual basis of $n$ functionals. Since a basis determines the dimension, $\dim V^* = n = \dim V$. $\blacksquare$
Prerequisites
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Prerequisites Graph
Interactive dependency map showing how this theorem builds on foundational concepts