# Linear maps. Proving its linear, and describing kernal.

• April 19th 2008, 04:39 AM
flawless
Linear maps. Proving its linear, and describing kernal.
Let c be an invertible nxn matrix and consider the map T:M(subscript)n->M(subscript)n defined by T(X)=CX-XC
(a) show that T is a linear map
(b) Describe kerT, hence show that T is never invertible.
Thanks in advance, you guys help so much. Even if something may be so simple, if someone doesnt take the time out to show you, you will never understand..Cheers
• April 19th 2008, 06:22 AM
flyingsquirrel
Hello

a) Let $\lambda\in\mathbb{K},\,X,\,Y\in\mathcal{M}_n(\math bb{K})$. You have to show that $T(\lambda X+Y)=\lambda T(X)+T(Y)$ which comes by developing $T(\lambda X+Y)=C(\lambda X+Y)-(\lambda X+Y)C=\ldots$

b) Hint : $T(X)$ is called the commutator of $X$ and $C$
• April 20th 2008, 12:46 AM
sk1001
look in page 54 of lecture notes