Let B be a fixed 2X2 matrix and define a function T:M_2x2 --->M_2x2 by T(A)= AB-BA. Prove or disprove that T is a linear operator.
In order to prove or disprove this statement, we need to know what does it mean that a mapping (or function) is linear. By definition, a transformation (or mapping) is linear if:
(i) for all in the domain of ; and
(ii) for all and all scalar .
So in order to prove is a linear operator, we have to show that it satisfies the above two properties. Let's verify the first property:
Let and be two matrices and is the given fixed matrix, by the given function and basic matrix operations, we have
which shows that satisfies the first property. Next you need to show for any given matrix and scalar . Can you pick up from here?
Roy