Using a orthonormal bases of $\mathbb R^4$ and $\mathbb R^6$ find a solution $Ax=z_1$ If you have orthonormal vectors $q_1,q_2,q_3,q_4$ of $\mathbb R^4$ and orthonormal vectors $z_1,z_2,z_3,z_4,z_5,z_6$ of $\mathbb R^6$ and matrix A is $A=z_1q_1^T+z_2q_2^T$.  a)Find a base and dimension of fundamental subspaces of  $A$  b) Using fact from a) find a solution $Ax=z_1$  For a)$ Im(A)=L(z_1,z_2), dimIm(A)=2, Im(A^T)=L(q_1,q_2), dim Im(A^T)=2, ker(A)=L(q_3,q_4) dimker(A)=2, ker(A^T)=L(z_3,z_4,z_5,z_6) ,dim N(A^T)=4.$  For b) $Ax=z_1$, If $z_1 \in Im(A)$ then this equation have solution, then $x=q_1$ since $(x_1,x_1)=1$ and $(x_2,x_1)=0$ so $Aq_1=(z_1q_1^Tq_1+z_2q_2^Tq_1)=z_1+0=z_1$ is this ok?
Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,...,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$. Let $q_1,q_2,q_3,q_4$ be orthonormal vectors in $\mathbb{R}^4$,$z_1,z_2,...,z_6$ orthonormal vectors in $\mathbb{R}^6$ and $A=z_1q_1^T + z_2q_2^T$.  a) Find the base and dimension of fundamental subspaces of matrix A  b) Using the results given by a) find the general solution for $Ax - z_1$.  Any hint would be appreciated.
What is the pronunciation of "the"? I read that the definite article is pronounced differently depending on the word that follows it.  Which is the exact pronunciation of the? Ubuntu 17.10 windows' menu in panel I installed the latest version of Ubuntu (17.10) , but the problem is, I can't find the menu bars of applications in the GNOME panel. For instance, the Chrome menu bar doesn't appear on the panel like in Ubuntu 17.04 and earlier versions. What are point releases in LTS versions? For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $ Slow math typing I have been using LaTeX for quite some time, but I find the math writing to be a bit frustrating. It seems to take me a relatively long time to code-write long math expressions, and sometimes it's hard to see where { start and end.  I am interested in writing as fast as possible.  Is code-typing each symbol really the fastest way? Is there any good program (in the spirit of Word's MathType) that enables producing math expressions by button clicking, that you recommend? How can I create launcher icon for PhpStorm? I have PhpStorm on my PC, to launch it I have to go to phpstorm/bin/ and launch phpstorm.sh, how can I pin this to launcher so that I dont have to go into folders every time? I have a table to store StudentID, SubjectID, ObtainedMarks. Now I have to create a pivot table to show studentID and obtainedMarks in different subject. Number of subject may differ. I could generate pivot table for fixed subjectID like    SELECT * FROM (SELECT StudentID, SubjectID, ObtainedMarks from [Exam].[ObtainedMarkEntry]) as InvoiceResult PIVOT (     sum(ObtainedMarks) for [SubjectID] in  ([1],[2],[3],[4]) )as finalResult   This works but problem is 1,2,3 is not static it should come from same table   So I did something like   declare @minvalue int, @maxvalue int declare @myArray varchar(max) set @MyArray=''  select RowID=IDENTITY(int, 1,1), SubjectID into #tempSubject from [Exam].[ObtainedMarkEntry] select * from #tempSubject set @minvalue = (select min(RowID) from #tempSubject) set @maxvalue = (select max(RowID) from #tempSubject) while (@minvalue&lt;=@maxvalue) begin     declare @sid nvarchar(10)     set @sid=(select SubjectID from #tempSubject where RowID=@minvalue)     set @myArray= @myArray + '[' + @sid + '], '      set @minvalue= @minvalue+1 end set @myArray= SUBSTRING(@myArray,0,len(@myArray)) drop table #tempSubject  declare @q as nvarchar(max) SELECT * FROM (SELECT StudentID, SubjectID, ObtainedMarks  from [Exam].[ObtainedMarkEntry]) as InvoiceResult PIVOT (     sum(ObtainedMarks) for [SubjectID] in  (@myArray) )as finalResult   Now I am getting error at in(@myArray) can't I assign variable in this section. Flag dialogue popup is stacked behind realtime-post-deleted-notification banners There is a little bug that becomes visible when a page contains realtime-post-deleted-notification banners (that's the CSS classname of these red banners appearing on answers which just got deleted, see screenshot below) and you try to open the flag dialogue above them:    Obviously, the dialogue appears below the banners, which it clearly should not do. It also has no input focus, trying to interact with the dialogue fails and either closes it (clicking outside of the red banners) or reloads the page (clicking on the red banners).  The cause for this is simply the z-index ordering in the site's CSS. Relevant CSS snippets:  .realtime-post-deleted-notification {     position: absolute;     text-align: center;     z-index: 100; }  .popup {     display: none;     padding: 15px;     position: absolute;     z-index: 10; }   So the fix for this minor bug would probably be to raise the z-index of the popup class to something greater than 100. How can I forward delete on iOS? Is there a way to delete text in front of the caret using the iOS software keyboard? The small wireless hardware keyboard accomplishes this with a fn+delete, so there might be a way to use multiple keys to accomplish the same task in iOS? Can somebody help me understanding the fouries series coefficients?  I know that if we have:  $$f(t) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$  And because of the $\sin(a+b)$ formula: $$ f(t) = \sin(2\pi nt + Ph) = \sin(2\pi nt )\cos(Ph_n) + \cos(2\pi nt )\sin(Ph_n)$$ Then:  $$\sum_{n=1}^N A_n \sin(2\pi nt + Ph) = \sum_{n=1}^N (A_n\sin(2\pi nt )\cos(Ph_n) + A_n\cos(2\pi nt )\sin(Ph_n))$$ And by definition: $$a_n = A_n\cos(Ph_n)$$ $$b_n = A_n\sin(Ph_n)$$ Then:  $$f(t) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph) = \sum_{n=1}^N (a_n\sin(2\pi nt ) + b_n\cos(2\pi nt ))$$ Where does the $A_n$ in the first formula come from? Could somebody explain me where does the $\frac{a_0}{2}$ in the formula below come from? $$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n \cos(nx) + b_n \sin(nx))$$  And the most important:  How can I proof the formulas for the $a_n$ and $b_n$ coefficients?  PLEASE, I'm searchinf for these answers for DAYS. Find all solutions to the functional equation $f(x) +f(x+y)=y+2 $ A cannonball is launched from a point 100 m above ground, at a speed of 20 m/s. This is modelled by the equation $h(t)=-4.9t^2 +20t +100$ The speed of sound (of the cannonball is) 380m/s If a person sitting directly underneath the cannonball runs as soon as they hear the sound do they escape the cannon? My main question is &quot;$y=-380x+100$&quot; the correct equation to model the cannon's sound? Proving a trig identity: $\frac{\sin(A + B)}{\sin(A - B)}=\frac{\tan A + \tan B}{\tan A - \tan B}$ Prove $ \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} >0 $ Prove that for $ 0&lt;x&lt; \pi $, $$ \quad S_n(x) = \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} &gt;0 \quad \forall n = 1,2,... $$  Having trouble with this problem. This is an olympiad-style question, so an answer that doesn't use calculus or analysis would be preferred. A possible approach is induction, but for this we need to find a function in terms of $n$ and $x$ so that we can actually use the inductive step. If anyone has any ideas they would be appreciated.  If you really want to go down the calculus route (at this point I don't mind), then $ S_n' (x) = \cos x + \cos 3x + ... +\cos((2n-1)x) $ , which you can find a closed form for, but I don't know how useful that is. Surface tension: the paper clip experiment In the paper clip experiment, the surface tension of water prevents the clip from falling, thus we can assume it exerts a force of $mg$ (weight of clip) upwards.   However, if you try to pull the clip out of water, surface tension opposes this motion also. If for simplicity we assume that the clip is a rectangle of length $l$ and breadth $b$, and the surface tension is $T$, then the liquid exerts a downward force of: $$T×2(l+b)$$  What is going on here? Why is there an inconsistency in the direction along which the force acts? A positive integer is divisible by $3$ iff 3 divides the sum of its digits I am having trouble proving the two following questions:    If $p|N$, $q|N$ and gcd(p,q)=1, then prove that $pq|N$ If $x$ is non zero positive integer number, then prove that $3|x$ if and only if 3 divides the sum of all digits of $x$.   For both questions I tried to use theorems of discrete mathematics, but I could not find the way to solve them. "Much of it is" or "Much of it are"? I'm writing an college-entrance essay and for me it's imperative that I understand and implement the correct one of these two expressions:     "Much of it is" or "Much of it are"   If you need context:     I'm having trouble with my writing. The reason I'm having trouble is that much of it is about....   Should there be an "is" or an "are"? Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$ Why no ooVoo on ubuntu? How can I use asymmetric encryption, such as RSA, to encrypt an arbitrary length of plaintext?