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: Nov 21, 2025
- \(T_1: M_3(\R)\rightarrow M_3(\R)\) 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
To determine which of the given maps from \(M_3(\R)\) to \(M_3(\R)\) are linear, we need to verify the conditions for linearity. A map \(T: V \rightarrow W\) is linear if for all vectors \(u, v \in V\) and all scalars \(c \in \R\), the following properties hold:
- \(T(u + v) = T(u) + T(v)\) (Additivity)
- \(T(cu) = cT(u)\) (Homogeneity)
We have the following transformations to consider:
-
\(T_1(M) = MP - PM\)
Let's check the linearity properties:- Additivity: \(T_1(M_1 + M_2) = (M_1 + M_2)P - P(M_1 + M_2)\) = \(M_1P + M_2P - (PM_1 + PM_2)\) = \((M_1P - PM_1) + (M_2P - PM_2)\) = \(T_1(M_1) + T_1(M_2)\).
- Homogeneity: \(T_1(cM) = (cM)P - P(cM) = c(MP) - c(PM) = c(MP - PM) = cT_1(M)\).
-
\(T_2(M) = M^2P - P^2M\)
Checking linearity:- Additivity: For matrices \(M_1, M_2\), \(T_2(M_1 + M_2)\) involves cross terms like \(M_1M_2P\) which would not split easily under linear mapping without additional conditions. Hence, it does not necessarily equal \(T_2(M_1) + T_2(M_2)\).
- Homogeneity: \(T_2(cM) = (cM)^2P - P^2(cM) = c^2M^2P - cP^2M\), which is not equal to \(cT_2(M)\) as the scalar factors differ.
-
\(T_3(M) = MP^2 + P^2M\)
Let's check the linearity:- Additivity: \(T_3(M_1 + M_2) = (M_1 + M_2)P^2 + P^2(M_1 + M_2)\) = \(M_1P^2 + M_2P^2 + P^2M_1 + P^2M_2\) = \((M_1P^2 + P^2M_1) + (M_2P^2 + P^2M_2)\) = \(T_3(M_1) + T_3(M_2)\).
- Homogeneity: \(T_3(cM) = (cM)P^2 + P^2(cM) = c(MP^2) + c(P^2M) = c(MP^2 + P^2M) = cT_3(M)\).
-
\(T_4(M) = MP^2 - PM^2\)
Testing for linearity:- Additivity: The expression \(T_4(M_1 + M_2) = (M_1 + M_2)P^2 - P(M_1 + M_2)^2\) leads to cross terms in the second part, similar to \(T_2\), implying non-linearity due to additional terms that do not cancel out simply.
- Homogeneity: \(T_4(cM) = (cM)P^2 - P(cM)^2 = c(MP^2) - c^2PM^2\), not equal to \(cT_4(M)\).
Therefore, the maps \(T_1\) and \(T_3\) are linear transformations from \(M_3(\R)\) to \(M_3(\R)\).
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