Question

Prove that if {𝒗1, 𝒗2, . . . , 𝒗𝑛 } is a basis for a vector space V and 𝒘1, 𝒘2, . . . , 𝒘𝑛 are vectors in a vector space W, not necessarily distinct, then there exists a linear transformation 𝑇 ∶ 𝑉 → 𝑊 such that 𝑇(𝒗1 ) = 𝒘1 , 𝑇(𝒗2 ) = 𝒘2 , . . ., 𝑇(𝒗𝑛 ) = 𝒘𝑛 .

Name: prove that if v1 v2 vn is a basis for a vector space v and w1 w2 wn are vectors in a vector space w not necessarily distinct then there exists a linear transformation t v w such that tv1 w1 tv2 w2 tvn
Uploaded: 2022-12-28T07:37:00-08:00
Duration: 3 min 19 s
Channel: Vincenzo Zaccaro
Description: prove that if v1 v2 vn is a basis for a vector space v and w1 w2 wn are vectors in a vector space w not necessarily distinct then there exists a linear transformation t v w such that tv1 w1 tv2 w2 tvn

          Prove that if {𝒗1, 𝒗2, . . . , 𝒗𝑛
} is a basis for a vector space V and 𝒘1, 𝒘2, . . . , 𝒘𝑛 are vectors 
in a vector space W, not necessarily distinct, then there exists a linear transformation 
𝑇 ∶ 𝑉 → 𝑊 such that
 𝑇(𝒗1
) = 𝒘1 , 𝑇(𝒗2
) = 𝒘2 , . . ., 𝑇(𝒗𝑛
) = 𝒘𝑛 .

Added by Yathursiya G.

Differential Equations and Linear Algebra

Stephen W. Goode, Scott A. Annin 4th Edition

Chapter 4

Instant Answer

Solved by Expert Vincenzo Zaccaro

12/28/2022

Step 1

. . , 𝒗𝑛 } is a basis for a vector space V and 𝒘1, 𝒘2, . . . , 𝒘𝑛 are vectors in a vector space W, not necessarily distinct. ** Show more…

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Prove that if {𝒗1, 𝒗2, . . . , 𝒗𝑛 } is a basis for a vector space V and 𝒘1, 𝒘2, . . . , 𝒘𝑛 are vectors in a vector space W, not necessarily distinct, then there exists a linear transformation 𝑇 ∶ 𝑉 → 𝑊 such that 𝑇(𝒗1 ) = 𝒘1 , 𝑇(𝒗2 ) = 𝒘2 , . . ., 𝑇(𝒗𝑛 ) = 𝒘𝑛 .