3 Suppose V1, …, Vm are vector spaces. Prove that ℒ(V1 ×…× Vm, W) and ℒ(V1, W) ×…×ℒ(Vm, W) are i

Name: 3 Suppose V1, …, Vm are vector spaces. Prove that ℒ(V1 ×…× Vm, W) and ℒ(V1, W) ×…×ℒ(Vm, W) are isomorphic vector spaces.
Uploaded: 2023-02-12T05:08:47-08:00
Duration: 3 min 21 s
Channel: Sri K
Description: 3 Suppose V1, …, Vm are vector spaces. Prove that ℒ(V1 ×…× Vm, W) and ℒ(V1, W) ×…×ℒ(Vm, W) are isomorphic vector spaces.

Step 1: Define a linear transformation $phi: mathcal{L}(V_1 times ... times V_m, W) rightar

Question

3 Suppose $V_1, \dots, V_m$ are vector spaces. Prove that $\mathcal{L}(V_1 \times \dots \times V_m, W)$ and $\mathcal{L}(V_1, W) \times \dots \times \mathcal{L}(V_m, W)$ are isomorphic vector spaces.

          3 Suppose $V_1, \dots, V_m$ are vector spaces. Prove that $\mathcal{L}(V_1 \times \dots \times V_m, W)$ and $\mathcal{L}(V_1, W) \times \dots \times \mathcal{L}(V_m, W)$ are isomorphic vector spaces.

3 Suppose V1, …, Vm are vector spaces. Prove that ℒ(V1 ×…× Vm, W) and ℒ(V1, W) ×…×ℒ(Vm, W) are isomorphic vector spaces.

Added by Dustin W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

02/12/2023

Step 1

.. \times V_m, W) \rightarrow \mathcal{L}(V_1, W) \times ... \times \mathcal{L}(V_m, W)$ by $\phi(T) = (T_1, ..., T_m)$, where $T_i: V_i \rightarrow W$ is defined by $T_i(v_i) = T(0, ..., 0, v_i, 0, ..., 0)$. Show more…

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3 Suppose $V_1,..., V_m$ are vector spaces. Prove that $\mathcal{L}(V_1 \times ... \times V_m, W)$ and $\mathcal{L}(V_1, W) \times ... \times \mathcal{L}(V_m, W)$ are isomorphic vector spaces.