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) $\displaystyle T$ is linear if:
(i) $\displaystyle T(u+v)=T(u)+T(v)$ for all $\displaystyle u,\;v$ in the domain of $\displaystyle T$; and
(ii) $\displaystyle T(\mathit{c}u)=\mathit{c}T(u)$ for all $\displaystyle u$ and all scalar $\displaystyle \mathit{c}$.
So in order to prove $\displaystyle T$ is a linear operator, we have to show that it satisfies the above two properties. Let's verify the first property:
Let $\displaystyle U$ and $\displaystyle V$ be two $\displaystyle 2\times 2$ matrices and $\displaystyle B$ is the given fixed $\displaystyle 2\times 2$ matrix, by the given function and basic matrix operations, we have
$\displaystyle T(U+V)=(U+V)B-B(U+V)=UB+VB-BU-BV$
$\displaystyle =UB-BU+VB-BV=T(V)+T(U)$
which shows that $\displaystyle T$ satisfies the first property. Next you need to show $\displaystyle T(\mathit{c}U)=\mathit{c}T(U)$ for any given $\displaystyle 2\times 2$ matrix $\displaystyle U$ and scalar $\displaystyle \mathit{c}$. Can you pick up from here?
Roy