Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question already has an answer here:

Suppose that $V$ is a finite dimensional vector space and $T$ is a linear transformation from $V$ to $V$. Prove that $T$ is a scalar multiple of the identity matrix iff $ST=TS$ for every linear transformation $S$ from $V$ to $V$.

share|cite|improve this question

marked as duplicate by rschwieb, 1015, Henry T. Horton, user1551, Andreas Caranti Feb 19 '13 at 18:06

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

One direction is trivial. For the other, take $\{v_1,...,v_n\}$ to be any basis for $V$, and consider the transformations $E_{i,j}:V\to V$ induced by $$v_k\mapsto\begin{cases} v_j & \text{if }k=i,\\0 & \text{otherwise.}\end{cases}$$ What can we conclude about $T$ from the fact that $E_{i,j}T=TE_{i,j}$ for all $i,j\in\{1,...,n\}$?

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.