Proof of a linear transformation property Suppose ϕ:X→Y is a map of sets and F is a field. Let ϕ⋅:F(Y)→F(X) be a map sending a function f∈F(Y) to a function ϕ⋅(f)∈F(X) given by ϕ⋅(f)(x)=f(ϕ(x)) for every x∈X. How can we prove that for a scalar λ∈F,λ⋅ϕ⋅(f)=ϕ⋅(λf)? Take it as axiomatic that ϕ⋅ is a ring homomorphism if needed.

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Proof of a linear transformation property  Suppose ϕ:X→Y is a map of sets and F is a field. Let ϕ⋅:F(Y)→F(X) be a map sending a function f∈F(Y) to a function ϕ⋅(f)∈F(X)  given by ϕ⋅(f)(x)=f(ϕ(x)) for every x∈X.  How can we prove that for a scalar λ∈F,λ⋅ϕ⋅(f)=ϕ⋅(λf)?  Take it as axiomatic that ϕ⋅ is a ring homomorphism if needed.

demiegiq1p · Accepted Answer

No need to know anything more about ϕ⋅.   By definition, we have  ϕ⋅(λf)(x)=(λf)(ϕ(x))=λf(ϕ(x))=λϕ⋅(x).  Note: by definition, the function λf is given by (λf)(y)=λf(y). This explains the middle step.

Proof of a linear transformation property Suppose \phi:X \rightarrow Y is a map of sets an

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