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3) Prove that (a) V₁ ^V₂ = 0 in ^²(V) iff every alternating bilinear map f: VxV→W has f(v₁, v₂) = 0 (b) Conclude that ^²(V)=0 is identically 0 (ie, f(v₁, v2) = 0 for all V₁, V2, V) (c) every alternating bilinear map f: VxV→W Prove that v₁ ^U₂ = v₁ ^ v₂ in ^²(V) iff every alternating bilinear map f: VxV→W

3) Prove that (a) V₁ ^V₂ = 0 in ^²(V) iff every alternating bilinear map f: VxV→W has f(v₁, v₂) = 0 (b) Conclude that ^²(V)=0 is identically 0 (ie, f(v₁, v2) = 0 for all V₁, V2, V) (c) every alternating bilinear map f: VxV→W Prove that v₁ ^U₂ = v₁ ^ v₂ in ^²(V) iff every alternating bilinear map f: VxV→W

Advanced Engineering Mathematics
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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3) Prove that (a) V₁ ^V₂ = 0 in ^²(V) iff every alternating bilinear map f: VxV→W has f(v₁, v₂) = 0 (b) Conclude that ^²(V) = 0 ⇒ every alternating bilinear map f: V × V →W is identically 0 (ie, ƒ(v₁, v₂) = 0 for all v₁, v2, € V) (c) Prove that v₁ ^ V₂ = v₁ ^ v½ in ^²(V) iff every alternating bilinear map f: VxV→W has the property that f(v₁, v₂) = f(v₁, v₂)
Transcribed Image Text:3) Prove that (a) V₁ ^V₂ = 0 in ^²(V) iff every alternating bilinear map f: VxV→W has f(v₁, v₂) = 0 (b) Conclude that ^²(V) = 0 ⇒ every alternating bilinear map f: V × V →W is identically 0 (ie, ƒ(v₁, v₂) = 0 for all v₁, v2, € V) (c) Prove that v₁ ^ V₂ = v₁ ^ v½ in ^²(V) iff every alternating bilinear map f: VxV→W has the property that f(v₁, v₂) = f(v₁, v₂)
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