Let $A \in Mat(2\times 2, \mathbb{Q})$ be a matrix with $AB = BA$ for all matrices $B \in Mat(2\times 2, \mathbb{Q})$.  Show that there exists a $\lambda \in \mathbb{Q}$ so that $A = \lambda E_2$.    Let $E_{ij}$ be the matrix with all entries $0$ except $e_{ij} = 1$.  $$ \begin{align} AE_{11} &amp;= E_{11}A \\ \left( \begin{array}{cc} a_{11} &amp; a_{12} \\ a_{21} &amp; a_{22} \\ \end{array} \right) \left( \begin{array}{cc} 1 &amp; 0 \\ 0 &amp; 0 \\ \end{array} \right) &amp;= \left( \begin{array}{cc} 1 &amp; 0 \\ 0 &amp; 0 \\ \end{array} \right) \left( \begin{array}{cc} a_{11} &amp; a_{12} \\ a_{21} &amp; a_{22} \\ \end{array} \right) \\ \left( \begin{array}{cc} a_{11} &amp; 0 \\ a_{21} &amp; 0 \\ \end{array} \right) &amp;= \left( \begin{array}{cc} a_{11} &amp; a_{12} \\ 0 &amp; 0 \\ \end{array} \right) \\ \end{align} $$  $\implies a_{12} = a_{21} = 0$  And then the same for the other three matrices $E_{12}, E_{21}, E_{22}$ …  I guess it's not the most efficient way of argueing or writing it down … ? Where's the trick to simplify this thingy?
The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study.  We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S \in {\cal L}(V), TS = ST$. Here, ${\cal L}(V)$ denotes the set of all linear operators over $V$.  One direction is easy to prove. If $T$ is a scalar multiple of the identity, then there exists a scalar $a$ such that $Tv = av$, $\forall v \in V$. Hence, given an arbitrary vector $w$, $$TS(w) = T(Sw) = a(Sw) = S(aw) = S(Tw) = ST(w)$$ where the third equality is possible because $S$ is a linear operator. Then, it follows that $TS = ST$, as required.  I am, however, at a loss as to how to tackle the other direction. I thought that a proof by contradiction,  ultimately constructing a linear operator $S$ for which $TS \neq ST$, might be the way to go, but haven't made much progress.   Thanks in advance!
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Things I've tried:   post suggests that I can use the undocumented code: operator, but when I do that, it just wraps the whole term in double quotes like .  As @terdon in Unix &amp; Linux chat has , using this  does yield some relevant results, but it doesn't show me  that is supposed to be there. edit: It appears that the Symbolhound query no longer returns that result.  Finally,  says that you can search for literal characters, but as you can see:    Quoted phrases are exact matches except for case-sensitivity, for example, you can search  or .  The second example (symbols) leads to 0 results.  So by now I assume that the search engine has changed (but the old documentation still lingers on this site). Is there a method for me to search for strings that include special (or is it literal?) characters? Why does the buffer take up whole screen? I am trying to create buffers around the Hurricane Sandy path I created in QGIS (points are lat/long).   I have both points from each day she was tracted and the line I created from it with Points2One.   When I go to make buffers around either the line or the points, the buffer takes up the whole screen. Its all just one color.   I have made buffers before but from data I did not create.   Right now all of my projections are set to WGS 84 EPSG 4326. How to tell whether two graphs are isomorphic? List of freely available physics books I'm trying to amass a list of physics books with open-source licenses, like Creative Commons, GPL, etc. The books can be about a particular field in physics or about physics  in general.  What are some freely available great physics books on the Internet? edit: I'm aware that there are tons of freely available lecture notes online. Still, it'd be nice to be able to know the best available free resources around.  As a starter:         Table of contents sorted by field (in alphabetical order): Accused of cheating on online exams I have 4GB RAM. 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