Question:
Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?
Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?
Updated On: Oct 27, 2024
- \(T_1: M_3(\R)\rightarrow M_3(\R)\) given by T1(M)=MP-PM for \(M\isin M_3(\R)\).
- \(T_2: M_3(\R)\rightarrow M_3(\R)\) given by T2(M)=M2P-P2M for \(M\isin M_3(\R)\).
- \(T_3: M_3(\R)\rightarrow M_3(\R)\) given by T3(M)=MP2+P2M for \(M\isin M_3(\R)\).
- \(T_4: M_3(\R)\rightarrow M_3(\R)\) given by T4(M)=MP2-PM2 for \(M\isin M_3(\R)\).
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The Correct Option is A, C
Solution and Explanation
The correct option is (A): \(T_1: M_3(\R)\rightarrow M_3(\R)\) given by T1(M)=MP-PM for \(M\isin M_3(\R)\). and (C): \(T_3: M_3(\R)\rightarrow M_3(\R)\) given by T3(M)=MP2+P2M for \(M\isin M_3(\R)\).
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