1. ## transpose

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.

2. Originally Posted by natewalker205
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) $T$ is linear if:
(i) $T(u+v)=T(u)+T(v)$ for all $u,\;v$ in the domain of $T$; and
(ii) $T(\mathit{c}u)=\mathit{c}T(u)$ for all $u$ and all scalar $\mathit{c}$.

So in order to prove $T$ is a linear operator, we have to show that it satisfies the above two properties. Let's verify the first property:
Let $U$ and $V$ be two $2\times 2$ matrices and $B$ is the given fixed $2\times 2$ matrix, by the given function and basic matrix operations, we have

$T(U+V)=(U+V)B-B(U+V)=UB+VB-BU-BV$
$=UB-BU+VB-BV=T(V)+T(U)$

which shows that $T$ satisfies the first property. Next you need to show $T(\mathit{c}U)=\mathit{c}T(U)$ for any given $2\times 2$ matrix $U$ and scalar $\mathit{c}$. Can you pick up from here?

Roy