Properties of dual linear transformation Let V,W finite dimensional vector spaces over the field F, and let T:V→W a linear transformation. Define T⋅:W⋅→V⋅ the dual linear transformation. i.e T⋅(ψ)=ψ∘T.

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Properties of dual linear transformation  Let V,W finite dimensional vector spaces over the field F, and let T:V→W a linear transformation.   Define T⋅:W⋅→V⋅ the dual linear transformation. i.e T⋅(ψ)=ψ∘T.

Walter Howe · Accepted Answer

Let A=(aij) be the matrix of T with respect to the bases {v1,⋯vn} of V and {w1,⋯wn} of W. y definition, this means that ϕ(vj)=∑i=1naijwi for j=1,⋅,n. To show that the matrix of T⋅ is AT, we need to show that T⋅(ψj)=∑i=1najiϕi. For each k=1,⋯,n, we have  T⋅(ψj)(vk)=ψj(T(vk))  ψj(∑i=1naikwi)  =∑i=1naikψj(wi)  =ajk  =∑i=1najiϕi(vk)  =(∑i=1najiϕi)(vk),  so T⋅(ψj) and ∑i=1n ajiϕi agree on a basis of V and are therefore equal.

Properties of dual linear transformation Let V,W finite dimensional vector spaces

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