📐 Dual Space Dimension

If $V$ is finite-dimensional, then $\\dim(V^*) = \\dim(V)$.

Proof: Let $\\{v_1, \\ldots, v_n\\}$ be a basis for $V$. Define $f_i: V \\to F$ by:

$$f_i(v_j) = \\delta_{ij} = \\begin{cases} 1 & i = j \\\\ 0 & i \\neq j \\end{cases}$$

**Claim:** $\\{f_1, \\ldots, f_n\\}$ is a basis for $V^*$.

**Spanning:** For any $f \\in V^*$, let $c_i = f(v_i)$. Then for any $v ...

From: Advanced Linear Algebra

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