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Question

Linear Algebra Question on Matrices

Let P be a fixed 3×3 matrix with entries in R\R. Which of the following maps from M3(R)M_3(\R) to M3(R)M_3(\R) is/are linear?

A

T1:M3(R)→M3(R)T_1: M_3(\R)\rightarrow M_3(\R) given by T1(M)=MP-PM for M∈M3(R)M\isin M_3(\R).

B

T2:M3(R)→M3(R)T_2: M_3(\R)\rightarrow M_3(\R) given by T2(M)=M2P-P2M for M∈M3(R)M\isin M_3(\R).

C

T3:M3(R)→M3(R)T_3: M_3(\R)\rightarrow M_3(\R) given by T3(M)=MP2+P2M for M∈M3(R)M\isin M_3(\R).

D

T4:M3(R)→M3(R)T_4: M_3(\R)\rightarrow M_3(\R) given by T4(M)=MP2-PM2 for M∈M3(R)M\isin M_3(\R).

Answer

T1:M3(R)→M3(R)T_1: M_3(\R)\rightarrow M_3(\R) given by T1(M)=MP-PM for M∈M3(R)M\isin M_3(\R).

Explanation

Solution

The correct option is (A): T1:M3(R)→M3(R)T_1: M_3(\R)\rightarrow M_3(\R) given by T1(M)=MP-PM for M∈M3(R)M\isin M_3(\R). and (C): T3:M3(R)→M3(R)T_3: M_3(\R)\rightarrow M_3(\R) given by T3(M)=MP2+P2M for M∈M3(R)M\isin M_3(\R).