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Prove the following generalization of Theorem 2.6: Let V and W be vector spaces over a
Chapter 2, Problem 34(choose chapter or problem)
Prove the following generalization of Theorem 2.6: Let V and W be vector spaces over a common field, and let ft be a basis for V. Then for any function f:ft>\N there exists exactly one linear transformation T: V -+ W such that T(x) = f(x) for all x ft.
Questions & Answers
QUESTION:
Prove the following generalization of Theorem 2.6: Let V and W be vector spaces over a common field, and let ft be a basis for V. Then for any function f:ft>\N there exists exactly one linear transformation T: V -+ W such that T(x) = f(x) for all x ft.
ANSWER:Step 1 of 2
Given that V and W be vector spaces over a common field, say F and 
Let 
We consider 
Since 
To prove that T is unique.
Let us consider an arbitrary element 
