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10 Nov 2019
If Tj : P(R) rightarrow P(R) be the linear operator sending a polynomial to its j-th derivative. Prove that T1,... ,Tn are linearly independent in L(P(R)). Let T : V rightarrow W be a linear transformation between F-vector space V and W of equal dimension. Prove that there exist basis beta of V and gamma of W such that [T]gamma beta is a diagonal matrix whose entries are 1 or 0.
If Tj : P(R) rightarrow P(R) be the linear operator sending a polynomial to its j-th derivative. Prove that T1,... ,Tn are linearly independent in L(P(R)). Let T : V rightarrow W be a linear transformation between F-vector space V and W of equal dimension. Prove that there exist basis beta of V and gamma of W such that [T]gamma beta is a diagonal matrix whose entries are 1 or 0.