Find linear transformation for matrices Question: Find all the linear transformations $f$ from $Mn(R)$ to $R$ such that $f(AB) = f(BA)$ for all matrices $A, B ∈ Mn(R)$. Doubt: I know what a linear transformation is, but I am not able to use the given linear transformation condition.
Linear maps $f: M_n \longrightarrow \mathbb{R}$ such that $f(AB) = f(BA)$ Let $M_n$ be the vector space of $n \times n$ matrices over $\mathbb{R}$. Find (with proof) all linear maps $f: M_n \longrightarrow \mathbb{R}$ such that $f(AB) = f(BA)$ for all matrices $A$ and $B$.   I know that the trace map is one such $f$. The determinant map satisfies $det(AB) = det(BA)$ -- but isn't a linear map, so it doesn't suffice.   Is there a constructive way to find all of the desired linear maps $f$, and show that there are no others ?   Any help would be appreciated. Thanks (=
What are the best JVM settings you have found for running Eclipse? How to let `dpkg -i` install dependencies for me? Say, I have foo-1.2.3.deb which depends on perl and python, however, running command:  dpkg -i ./foo-1.2.3.deb   won't install these dependencies. So I must apt-get install perl python by hand.  How to make dpkg -i install these dependencies for me automatically? Find the value of $\binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots \binom{2000}{2000}$ Find the value of $\binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots \binom{2000}{2000}$     I've seen many complex proofs. I am looking for an elementary proof. I know the fact that $\binom{2000}{0} + \binom{2000}{1} + \binom{2000}{2} + \cdots \binom{2000}{2000} = 2^{2000}$. This may help here. Is the consciousness currently in oneself the same the the consciousness that used to be or in the future will be in oneself? People seem to believe that one's consciousness is a "different" consciousness than that of others, but the "same" consciousness as the one that has been in one's body in the past and the one that will be in one's body in the future. How likely is this to be true and why?   Also, I have almost no knowledge about neuroscience, so please refrain from using neuroscience jargon if feasible. Is there a cross-browser onload event when clicking the back button? For all major browsers (except IE), the JavaScript onload event doesn’t fire when the page loads as a result of a back button operation — it only fires when the page is first loaded.  Can someone point me at some sample cross-browser code (Firefox, Opera, Safari, IE, …) that solves this problem? I’m familiar with Firefox’s pageshow event but unfortunately neither Opera nor Safari implement this. Question about repeating decimal? For simple fraction, we can easily convert it to repeating decimal by calculator. Ex. $\frac 1 3 = 0.33333\ldots$, $\frac 1 7=0.(142857),\ldots$ But some fraction fraction like $10/29, 1/97,...$ The repeating part of them are too long, so it can't fully show on the calculator. So is there any algorithm  to find the repeating part for that fraction? How to keep processes running after ending ssh session? How can a point-particle have properties? I have trouble imagining how two point-particles can have different properties.  And how can finite mass, and finite information (ie spin, electric charge etc.) be stored in 0 volume?  Not only that, but it can also detect all fields without having any structure. Maybe it can check curvature of spacetime to account for gravity, but how can a point contain the information of what the other fields-vectors are? This seems to mean that also the information/volume in space is infinite.  Mathematically, a point cant have any intrinsic structure, so how does physics which is a mathematical theory explain this? I have asked this question before ( and  )  My problem is that my Self Encrypted Disk (SED) have to NOT spin down even on suspend!  I have tried , but it did not work and the disk spin downs anyway !  The problem is that if the disk spins down, because it is on suspend, it does not show the "de encryption" prompt, and boots back up on RAM, but crashed due to IO error on the disk! (that's what the disk is suposed to do !)  Anyone on how to NOT (as in NEVER) spin down a HDD ? (yes this goes against all logic but this particular case)   Thanks a lot !  Would that be useful ?   sudo hdparm -I /dev/sda [sudo] password for xqua:   /dev/sda:  ATA device, with non-removable media     Model Number:       ST500LT015-9WU142                            Serial Number:      W0VBZQQ7     Firmware Revision:  0001SDM7     Transport:          Serial, ATA8-AST, SATA 1.0a, SATA II Extensions, SATA Rev 2.5, SATA Rev 2.6 Standards:     Used: unknown (minor revision code 0x0029)      Supported: 8 7 6 5      Likely used: 8 Configuration:     Logical     max current     cylinders   16383   16383     heads       16  16     sectors/track   63  63     --     CHS current addressable sectors:   16514064     LBA    user addressable sectors:  268435455     LBA48  user addressable sectors:  976773168     Logical  Sector size:                   512 bytes     Physical Sector size:                  4096 bytes     Logical Sector-0 offset:                  0 bytes     device size with M = 1024*1024:      476940 MBytes     device size with M = 1000*1000:      500107 MBytes (500 GB)     cache/buffer size  = 16384 KBytes     Form Factor: 2.5 inch     Nominal Media Rotation Rate: 5400 Capabilities:     LBA, IORDY(can be disabled)     Queue depth: 32     Standby timer values: spec'd by Standard, no device specific minimum     R/W multiple sector transfer: Max = 1   Current = 1     Advanced power management level: 254     Recommended acoustic management value: 208, current value: 0     DMA: mdma0 mdma1 mdma2 udma0 udma1 udma2 udma3 udma4 udma5 *udma6           Cycle time: min=120ns recommended=120ns     PIO: pio0 pio1 pio2 pio3 pio4           Cycle time: no flow control=120ns  IORDY flow control=120ns Commands/features:     Enabled Supported:        *    SMART feature set        *    Power Management feature set        *    Write cache        *    Look-ahead        *    Host Protected Area feature set        *    WRITE_BUFFER command        *    READ_BUFFER command        *    NOP cmd        *    DOWNLOAD_MICROCODE        *    Advanced Power Management feature set             Power-Up In Standby feature set        *    SET_FEATURES required to spinup after power up             SET_MAX security extension        *    48-bit Address feature set        *    Device Configuration Overlay feature set        *    Mandatory FLUSH_CACHE        *    FLUSH_CACHE_EXT        *    SMART error logging        *    SMART self-test        *    General Purpose Logging feature set        *    WRITE_{DMA|MULTIPLE}_FUA_EXT        *    64-bit World wide name        *    IDLE_IMMEDIATE with UNLOAD        *    Write-Read-Verify feature set        *    WRITE_UNCORRECTABLE_EXT command        *    {READ,WRITE}_DMA_EXT_GPL commands        *    Segmented DOWNLOAD_MICROCODE        *    Gen1 signaling speed (1.5Gb/s)        *    Gen2 signaling speed (3.0Gb/s)        *    Native Command Queueing (NCQ)        *    Host-initiated interface power management        *    Phy event counters        *    Idle-Unload when NCQ is active             Device-initiated interface power management        *    Software settings preservation        *    SMART Command Transport (SCT) feature set        *    SCT Write Same (AC2)        *    SCT Features Control (AC4)        *    SCT Data Tables (AC5)             unknown 206[12] (vendor specific)        *    reserved 69[4]        *    reserved 69[7] Logical Unit WWN Device Identifier: 5000c5006a036439     NAA     : 5     IEEE OUI    : 000c50     Unique ID   : 06a036439 Checksum: correct How are "Suggested Tags" chosen? After you've typed in a question, you get suggested tags that appear that look like this:        It appears that you need a minimum amount of characters ():     you must have entered a title (any length) and a minimum body length of 220 characters.   But other than that, how are the tags chosen? Is it:   A match of the title/body with other questions (taking the tags from those questions) A match of the title/body with tag excerpts (taking tags with overlap) A match of the title/body with tag wikis (taking tags with overlap) All of the Above None of the Above Other (please explain)   Some additional info from :  The suggested tags seem to only pop up on meta.so (they do not work on regular Stack Overflow, or on other SE sites).  While we're on the topic, why isn't this feature implemented network-wide? Find equation of a plane that passes through point and contains the intersection line of 2 other planes Find equation of a plane that passes through point P $(-1,4,2)$ that contains the intersection line of the planes  $$\begin{align*} 4x-y+z-2&amp;=0\\ 2x+y-2z-3&amp;=0 \end{align*}$$  Attempt: I found the the direction vector of the intersection line by taking the cross product of vectors normal to the known planes. I got $\langle 1,10,6\rangle$. Now, I need to find a vector normal to the plane I am looking for. To do that I need one more point on that plane. So how do I proceed? If $f:D\to D’$ is analytic and $u: D'\to R$ is harmonic then the composition of $u$ and $f$ is harmonic in $D$ If $f:D\to D’$ is analytic and $u: D'\to R$ is harmonic then the composition of $u$ and $f$ is harmonic in $D$.  How can I show that the above statement is true/false? Can anyone help me? How to prove useful property of logarithm of generating functional in QFT? How to prove that $\ln(Z(J))$ generates only? I can't find the proof of this statement, and have only met its demonstrations for case of 2- and 4-point. Is there a drupal.org API for retrieving the list of available modules, a description of them and available versions? Subject-Verb Agreement VMware Linux Server -- how can you tell if you are a vm or real hardware? Is it possible/sensible to add a suspension fork to an old mountain bike? I have a ~1993 Giant mountain bike.  It's not great, but I am a very new and casual rider.    The bike has a solid fork and it makes for the predictable rough trail ride.  The trails are generally fairly modest (I live in North Texas, a 100' change in altitude makes our ears pop here) and the surface is generally dirt (very few rocky areas).  So, I am thinking that it would be very nice to have a suspension bike.  And seeing that a modest fork can be purchased for a modest price, I am wondering if it makes sense to add a fork to this bike.   asked about going the other direction, and there were some cautionary words about changing steering geometry.  So, on balance, is this something that I would be best off to avoid?  I suspect that if I have to replace a bunch of steering parts, the cost will quickly approach that of a new low end bike. SELECT DISTINCT on multiple columns Supposing we have a table with four columns (a,b,c,d) of the same data type.  Is it possible to select all distinct values within the data in the columns and return them as a single column or do I have to create a function to achieve this? Can I make a custom node from a group and have that available in other projects? I've found some neat combinations of nodes to create certain effects, so I'd like to make them a group and then have that node available on my list of available nodes, even when I reload the startup file. How would I do that? This is a  about DNS server administration.   I have one hundred or so domains. All of these domains need to be configured identically, but it seems like a huge waste of time to have to configure a new zone and/or zonefile for every one of these domains. There has to be a better way to automate this!  I think I'm on to something...if I create a zone called ., or use some other feature in my DNS software to always return a specific IP when an A record is requested, this seems to get me pretty close to my desired end result. My server is responding authoritatively to the requests and it's so much easier to manage!  This was working great until nameserver validation software started checking these domains. I figured out that I can make most of the errors go away by adding NS records, but my software won't let me put more than one SOA record in the same zone file.  How do I work around this multiple SOA record problem?