Dimension of Preimage under Evaluation Isomorphism of Annihilator on Algebraic Dual
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Let $G$ be an $n$-dimensional vector space over a field.
Let $J: G \to G^{**}$ be the evaluation isomorphism.
Let $G^*$ be the algebraic dual of $G$.
Let $G^{**}$ be the algebraic dual of $G^*$.
Let $N$ be a $p$-dimensional subspace of $G^*$.
Then:
- $\map {J^\gets} {N^\circ}$ is an $\paren {n - p}$-dimensional subspace of $G$
where
Proof
We have that $N$ is a $p$-dimensional subspace of $G^*$.
From Dimension of Annihilator on Algebraic Dual we have that $N^\circ$ has dimension $n - p$.
From Inverse Evaluation Isomorphism of Annihilator we also have that $J^{-1} \sqbrk {N^\circ}$ also has dimension $n - p$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Theorem $28.10$