Prove that if $T,U: V \rightarrow V$ are linear operators on an inner product space, then $(TU)^* = U^*T^*$ So, the $^*$ represents taking the conjugate and transpose. I've been tinkering around with this by writing $T$ and $U$ as some sort of matrix and work out the calculations but I keep getting lost. is there a standard way to show this? Thank you!
Let be $V$ a vector space over the complex numbers with inner product. Let be $T$ and $S$ linear operators in $V$ with adjoint operators $T^*$, and $S^*$ respectively. I know that it satisfies this: $\left&lt; T(x),y \right&gt;=\left&lt; x,T^{*}(y) \right&gt;$ $\left&lt; S(x),y \right&gt;=\left&lt; x,S^{*}(y) \right&gt;$    $\forall x,y \in V$ But, How can I prove that the adjoint operator of $ST$ is $(ST)^{*}$? And how can I  prove that $(ST)^{*}=T^{*}S^{*}$?
Translation help review queue? Looking for the actual reason of refraction explained precisely without analogies I am making a fudge recipe that calls for 1cup whole milk and 5 cups sugar. It also uses 2 sticks of butter and 25 marshmallows. I'm wondering if I can omit the milk and cut back on the sugar by using sweetened condensed milk. I want to make a persistent live Ubuntu USB drive with more than 4GB of storage.   I know there is an option of installing directly on the drive but I don't want to do this as it will restrict the drive from working on different computers.   One option is to make an additional partition with extra space but if I do this how can I get newly installed software to be stored on this partition? multi-layer board copper pour I have a 6-layer board where the internal 4 planes are +15,GND,VCC,-15.  I was wondering if there was any advantage to doing a copper pour on the top and bottom layers?  I would probably leave them floating as I don't want to use micro-vias to say tie it to GND?    Is this actually a bad idea?  I.e. floating copper=antenna.  Would it be just as acceptable to have 4 layer board with the top layer having copper pour to VCC and the bottom having a pour to GND and keep the two internals as +-15?  Note that this is for a fairly low speed circuit which has some analog and digital parts. How do I install a fingerprint reader on Lenovo ThinkPad? How can I set the processor affinity of a process on Linux? How can I set the processor affinity of a process on Linux? Use fixed keyboard for a given app I have a very big file 4GB and when I try to read it my computer hangs. So I want to read it piece by piece and after processing each piece store the processed piece into another file and read next piece.  Is there any method to yield these pieces ?  I would love to have a lazy method. I want to add a check constraint on a column if entered value for column is greater than 3 than it should be saved as 1 else entered value. For that scenario I am using below query but on insertion of data it shows an error message.  create table tblTestCheckConstraint  ( id int, NewColumn int )   alter table tblTestCheckConstraint add constraint chk_tblTestCheckConstraint_NewColumn1 CHECK   (     CASE         WHEN NewColumn &gt;4 THEN 1         ELSE NewColumn     END = 1 )   Insert Statement:   insert into tblTestCheckConstraint values ( 1,5) insert into tblTestCheckConstraint values ( 1,2)   Error Message:   Msg 547, Level 16, State 0, Line 1 The INSERT statement conflicted with the CHECK constraint "chk_tblTestCheckConstraint_NewColumn1". The conflict occurred in database "DBName" , table "dbo.tblTestCheckConstraint", column 'NewColumn'. The statement has been terminated. I have a large amount of duplicate records (25.000) in an even larger table (250.000 records). I've done the detection using the reporting function. The criteria I used to detect the duplicates were 3 fields (unique account identifier, unique user identifier and date). Based on a list of criteria, I would like to either merge or delete these records. I've got a handle on the logic associated with the merging and the deleting but what I'm having trouble with is defining the duplicate detection process.   I'm looking at either creating a custom object to take care of the detection or dumping the records into an SQL database and taking care of it offline. Any thoughts on which one of the two is the best approach?  Edit:  After reading the first comment, I think the more appropriate question would be "How do I approach the problem in SOQL".  The object I'm working with is the Event object and I'm really only looking at the standard fields, AccountId, OwnerId and ActivityDate. From what I've found online, there's no easy way to find records that have the same value for these three fields. I think it could be accomplished by nesting SOQL queries but I'd like to use a different approach if at all possible. Screen goes white when posting a comment Since iOS 11.2 all the way through 12.2 beta 4, if you post or discard a comment on any post on any SE community, the screen will turn white. This has been ongoing continuously for a year now, and it affects both the SE app and the SO app.   You can go back to the previous screen and open the question again, and everything appears as normal. You can also rotate the screen to get the text to come back. Using the arrows to navigate between the question and answers updates the HUD, but the screen remains white unless one of the above is done to get the text back.        UPDATE: After posting anything (comment, answer), the screen goes all white except for the sidebar. Perhaps it is intentional, but I prefer last week’s behavior of going back to the content. Not major, since the sidebar icons still work. I'm currently taking Quantum Mechanics in school and I came across a table displaying the following in the given order:   $E_n (\text{energy levels}), g_n (\text{eigen functions})$   $n \ l, [m]$  $E_1, 1$   $1s, [0]$  $E_2, 4$   $2s, [0]$ $2p, [-1,0,1]$  $E_3, 9$   $3s, [0]$ $3p, [-1,0,1]$ $3d, [-2,-1,0,1,2]$    Now, as I understand $n$ defines the size of the orbital, $l$ the shape and $m$ the orientation.  My question is, how does the atoms shell model fit into this if one can have two electrons per orbital? Are $1s$ the same as the orbital $2s$, but just renamed for the sake of higher energy levels? Or do two similarly looking orbits exist for different energy levels? For example two "layers" of spherical electron clouds for an atom with $4$ electrons.  I just can't see how $2$, $8$, $18$ becomes $2$, $8$, $8$ as we were taught in Physics at high school? Am I missing something? "The queen of England's crown" or "The queen's of England crown"? Possible Duplicate:          What is the correct way of these two sentences?   The queen of England's crown The queen's of England crown   Strictly linguistically, sentence 2 should be the correct one, since the crown belongs to the queen and not to England but it sounds really awkward to pronounce. I have the following problem:  Let $a, b \in\mathbb{Z}$. Show that $\,\{ ax + by\ :\ x, y \in \mathbb{Z}\} = \{ n \gcd(a,b)\ :\ n\in \mathbb{Z} \}$   I understand that the Bezout's lemma says that $gcd(a,b) = ax +by$, so Im not really how you would go about proving the above, it doesn't really make sense to me. Any help is appreciated1 Solve the recurrence relation:$ T(n) = \sqrt{n} T \left(\sqrt n \right) + n$ $$T(n) = \sqrt{n} T \left(\sqrt n \right) + n$$  Master method does not apply here. Recursion tree goes a long way. Iteration method would be preferable.  The answer is $Θ (n \log \log n)$.  Can anyone arrive at the solution. Using ST_Union in PostGIS I have a spatial table (polygons,polylines) in a database. These are digitized by multiple people. For example, a state is divided into multiple Grids and given to multiple people to digitize using QGIS; which is connected to a postgres database. Now, I have to merge all those polygons and polylines. I don't know how to do this via ST_Union. I Used the following snippet to merge the polygons and polylines CREATE TABLE schema_name.polygon_un2 AS SELECT upg.un_geom,npg.Column1,npg.Column2,npg.Column3,npg.Column4,npg.Column5  FROM (  SELECT st_union(the_geom) As un_geom,(   SELECT oid FROM schema_name.polygon    WHERE oid = 100748191   ) FROM schema_name.polygon WHERE oid IN (100748191,100763519)  ) upg  INNER JOIN schema_name.polygon npg ON npg.oid = upg.oid;   DELETE FROM schema_name.polygon WHERE oid IN (100748191,100763519);  I have the interesecting polygon oid in a Seperate Table, I want to do this process in an update query and that should be a dynamic plpgsql function.  Green : Grid , Other : Polygon But This snippet did not work !! The snippet which i expect is  It should create a new row for the merged polygons and polylines (eg oid - 1 and 2 merged to oid 3).  The polygons which are not merged (eg 1 and 2 should be deleted) Is there a name for when the same word is used at the beginning and end of different phrases/sentences? Is there a name for when one phrase ends with a word that starts the next phrase. For instance:     life is a peach and cream   Or     sunscreen in the eye for detail   Or (from Dave Eggers' Heartbreaking Work of Staggering Genius)     JFKFC Paging Lucene's search results I am using Lucene to show search results in a web application.I am also custom paging for showing the same. Search results could vary from 5000 to 10000 or more. Can someone please tell me the best strategy for paging and caching the search results? Why is a linear transformation called linear?