2. Let a = {vi, ... , vin} be a finite ordered basis for V. For i define fi : V — F byMaivi ± • • •...

Question

2. Let a = {vi, ... , vin} be a finite ordered basis for V. For i define fi : V —> F by
Maivi ± • • • + anvn) = ai.
Then 0 { fl, ... , fn} is a basis for the vector space £(V, F) (you don't need to prove this). Let T : V —> V be a linear operator. Prove that
[T*10 ( [Tiarr)
(the transpose of the matrix [T]a.)

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Robert · Accepted Answer

Solutions
October 13, 2017
1. Let α = {v1, . . . , vn} be a finite ordered basis for V . For i = 1, . . . , n, define fi : V → F by
fi(a1v1 + · · ·+ anvn) = ai.
Then β = {f1, . . . , fn} is a basis for the vector space L(V, F ). Let T : V → V be a linear
operator. Prove that
[T ∗]β = ([T ]α)
tr.
Note: Here,

2. Let a = {vi, ... , vin} be a finite ordered basis for V. For i define fi : V —> F by Maivi ± • • • + anvn) = ai. Then 0 { fl, ... , fn} is a basis for the vector space £(V, F) (you don't need to...

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