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Linear mapping/Matrix to bases/Correspondence/Fact/Proof

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Proof

We show that both compositions are the identity. We start with a matrix and consider the matrix

Two matrices are equal, when the entries coincide for every index pair . We have


Now, let be a linear mapping, we consider

Two linear mappings coincide, due to fact, when they have the same values on the basis . We have

Due to the definition, the coefficient is the -th coordinate of with respect to the basis . Hence, this sum equals .