3) Let V have a basis {v1, ..., vn}. Let V* := L(V, F). Prove that the map D : V → V* 0 i+j and extending linearly i = j defined by (v;)(vi) := an isomorphism, thus V - V*.
3) Let V have a basis {v1, ..., vn}. Let V* := L(V, F). Prove that the map D : V → V* 0 i+j and extending linearly i = j defined by (v;)(vi) := an isomorphism, thus V - V*.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
18
![(18) Let \( V \) have a basis \(\{v_1, \ldots, v_n\}\). Let \( V^* := \mathcal{L}(V, \mathbb{F}) \). Prove that the map \(\Phi : V \to V^*\) defined by
\[
\Phi(v_j)(v_i) :=
\begin{cases}
0 & i \neq j \\
1 & i = j
\end{cases}
\]
and extending linearly is an isomorphism, thus \( V \simeq V^* \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F604590fe-432d-4635-a8d8-4a91503e7bd4%2F2ac8ceed-76cb-4163-abcc-e7544284988d%2Flhca8zv_processed.png&w=3840&q=75)
Transcribed Image Text:(18) Let \( V \) have a basis \(\{v_1, \ldots, v_n\}\). Let \( V^* := \mathcal{L}(V, \mathbb{F}) \). Prove that the map \(\Phi : V \to V^*\) defined by
\[
\Phi(v_j)(v_i) :=
\begin{cases}
0 & i \neq j \\
1 & i = j
\end{cases}
\]
and extending linearly is an isomorphism, thus \( V \simeq V^* \).
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