Recall that the direct sum V₁ V₂ is defined to be the space of pairs of vectors (v₁, v2) such that V1 € V₁, V2 € V₂ where the vector space structure is defined by the componentwise addition and simultaneous scalar multiplication. (1) Show that there is an isomorphism U ® (V₁ © V₂) ≈ (U ® V₁ ) + (U ® V₂). (2) Show that (V₁ © V₂)* ≈ V₁* © V₂. (3)* Show that (V₁ ® V₂) * = V₁ V₂.
Recall that the direct sum V₁ V₂ is defined to be the space of pairs of vectors (v₁, v2) such that V1 € V₁, V2 € V₂ where the vector space structure is defined by the componentwise addition and simultaneous scalar multiplication. (1) Show that there is an isomorphism U ® (V₁ © V₂) ≈ (U ® V₁ ) + (U ® V₂). (2) Show that (V₁ © V₂)* ≈ V₁* © V₂. (3)* Show that (V₁ ® V₂) * = V₁ V₂.
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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Transcribed Image Text:Recall that the direct sum V₁ V2 is defined to be the space of pairs of vectors (v₁, V₂) such
that v₁ € V₁, V2 € V₂ where the vector space structure is defined by the componentwise addition
and simultaneous scalar multiplication.
(1) Show that there is an isomorphism
U ® (V₁ © V₂) ≈ (U ® V₁ ) ℗ (U ® V₂).
(2) Show that (V₁ © V₂2)* ≈ V₁* © V₂* .
(3)* Show that (V₁ ® V ₂)* ≈ V₁ V₂.
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