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Use the universal property of the tensor product to show that: given line maps T₁: V₁ → W₁ and T₂ : V₂ → W₂ we get a well defined linear map T₁ T2: V₁ V₂ → W₁ 0 W₂ with the property that (T₁ ® T₂) (v₁ 0 V₂) = T₁(v₁) | T₂(v₂) for all v₁ € V₁, V₂ € V₂

Use the universal property of the tensor product to show that: given line maps T₁: V₁ → W₁ and T₂ : V₂ → W₂ we get a well defined linear map T₁ T2: V₁ V₂ → W₁ 0 W₂ with the property that (T₁ ® T₂) (v₁ 0 V₂) = T₁(v₁) | T₂(v₂) for all v₁ € V₁, V₂ € V₂

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 21EQ
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how can i use the universal property of the tensor product to solve this? Should I construct a map? bilinear map? i need some explanation with this. Thank you

Use the universal property of the tensor product to show that: given linear maps T₁: V₁ → W₁ and T2 : V₂ → W₂ we get a well defined linear map T₁ T2 : V₁0 V₂ → W₁ 0 W₂ with the property that (T₁ T₂) (V₁ 0 V₂) = T₁ (v₁) ❀ T₂(v₂) for all v₁ € V₁, V₂ € V₂
Transcribed Image Text:Use the universal property of the tensor product to show that: given linear maps T₁: V₁ → W₁ and T2 : V₂ → W₂ we get a well defined linear map T₁ T2 : V₁0 V₂ → W₁ 0 W₂ with the property that (T₁ T₂) (V₁ 0 V₂) = T₁ (v₁) ❀ T₂(v₂) for all v₁ € V₁, V₂ € V₂
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