I write like following my prove, and I need to know if it's correct: If P is an orthogonal matrix, then $$PP^t = I = P^tP$$ Therefore, $$\det(P^tP) = \det (I) \implies \det(P) \det (P^t) \implies \det (P) \det (P) = [\det (P)]^2$$   So $$ [\det(P)]^2 = 1 \implies \det (P) = \pm \sqrt 1 $$  So  $$ \det (P) = \pm 1$$ I prove it writing this, I would like to know if this proof is correct.
I have a question on my assignment asking me to prove that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$. What I did so far is:  We know that $A\cdot A^t = I$ (since it is orthogonal), and that would mean that if $A$ is a matrix with elements $a,b,c,d$ then :  \begin{cases}a^2 + b^2 = 1\\ ac + bd = 0\\ c^2 + d^2 = 1\end{cases}  which implies that \begin{align}\det(A)&amp;= ad - bc\\ &amp;= -\frac{bd^2}{c} - cb\\ &amp;= -\frac{bd^2}{c} - \frac{c^2b}{c}\\ &amp;= -\frac{bd^2 + bc^2}{c}\\ &amp;= -\frac{b(d^2 + c^2)}{c}\\ &amp;= -\frac{b(1)}{c}\\ &amp;= -b/c\end{align}  I might be doing something wrong here, could someone please help me out understand this?
Show that    $$\frac{\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}}{\binom{M}{n}}=\frac{k}{M},$$ where $M-k&gt;n$ and $k&gt;n$   From Vandermonde's identity, I get $\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}=\binom{M}{n}$.   But what about the sum of the product in the numerator? Any hint will be helpful. How to mount a NTFS partition in /etc/fstab? I've posted a lot to StackOverflow, and I've noticed something kind of irritating with the formatting in writing questions and comments. If I need to put something inside a &lt; and a >, without spaces in the middle (like right there), it will think it's code and not show up, like this   (obviously it won't come up what I wrote there, or else this wouldn't be an issue, but if you open up the question to edit you can see what's written there.)  If I want it to show up, right now I have to add an extra set of brackets inside of the originals with a junk variable, like right &lt;here> (again, open in edit window to see).  This is really annoying when trying to show variables like lists that require those brackets, and I feel like the junk variables could confuse people looking at the edits. Is there a way to get around this formatting? Thanks! Viewshed Analysis With GDAL I am looking to perform a simple (one observer location) viewshed analysis on a standard DEM using GDAL (either python or .NET bindings).  I have searched, but haven't found any readily available implementation.  Can anyone point me in the right direction? Override route helper methods Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K[x]$ imply relatively prime in $L[x]$? $K,L$ are fields, $K\subseteq L$. $f,g \in K[x]$. Suppose that $f,g$ are relatively prime as elements of $K[x]$. Prove they remain relatively prime  in $L[x]$.  I've tried everything I can think of. I feel like working with the contrapositive may be helpful but that's just a feeling. Setting READ UNCOMMITTED when reading DMVs I've seen several people call SET TRANSACTION ISOLATION LEVEL READ UNCOMMITTED before reading system DMVs. Is there ever any reason to do this, assuming you aren't mixing calls to DMVs and tables in the same transaction? first of all thank you for answering.  I have a 120GB SSD and a 1TB HDD and I want to dual boot with Windows 10. I tried with this configuration:  SSD  / (ext4 20GB) /usr (ext4 15GB) /boot/efi (ext4 250MB)  HDD  /var (ext4 3GB) /tmp (ext4 8GB) /home (ext4 50GB) swap (none 24GB) (RAM 16GB)  I don't know if I am doing OK, as I tried to install it but installation gets stuck at "Saving installed packages..." and it doesn't continue. I wonder if it could be cause I didn't  partitioned as I had to. I don't find any post where this happens so I ask here.  Thanks! In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I wrote below is valid?  Let $S$ be an infinite subset of a countable set $T$. Since $T$ is countable, there exists a bijection $f: \mathbb{N} \rightarrow T$. And $S \subseteq T = \{ f(n) \ | \ n \in \mathbb{N} \}$. Let $n_{1}$ be the smallest positive integer such that $f(n_{1}) \in S$. And continue where $n_{k}$ is the smallest positive integer greater than $n_{k-1}$ such that $f(n_{k}) \in S$. And because $S$ is infinite, we continue forever. Now consider the function $\beta : \mathbb{N} \rightarrow S$ which sends $k \rightarrow f(n_{k})$.   So in order for $S$ to be countable, $\beta$ would have to be a bijection. $\beta$ is injective since if $f(n_{k}) = f(n_{j})$, then $n_{k} = n_{j}$ because $f$ is injective. We also can conclude $k = j$ because the various $n_{i}$ chosen were strictly increasing.   Edit: $\beta$ also needs to be surjective. So for every $r \in S$ there exists a $q \in \mathbb{N}$ such that $\beta(q) = f(n_{q}) = r$. We know that $f^{-1}(r) \in \{n_{1}, n_{2}, ..., n_{k} ... \}$, so we can let $f^{-1}(r) = n_{d}$. Thus $\beta(d) = f(n_{d}) = r$. The rule says if the order of two actions/events is clear then the past perfect is not necessary, i.e. you can use past simple. So if the sentence has before or after then the order is clear and therefore the rule applies.  says the first sentence is incorrect. Is he/she right? If so, how do you explain what appears, to me, to be contradictory?     If the Past Perfect is not referring to an action at a specific time,   Past Perfect is not optional. Compare the examples below. Here Past   Perfect is referring to a lack of experience rather than an action at   a specific time. For this reason, Simple Past cannot be used.      Examples:      She never saw a bear before she moved to Alaska. Not Correct      She had never seen a bear before she moved to Alaska. Correct   They say "a lack of experience rather than an action". Saying that just does not solve the problem I have which is the past simple fits the rule of order of events. Not seeing is a negative event with the verb "to see".  I hope the answer won't be this is how we speak or this is how English works or since when languages are logical! How can I automatically log out the guest account after a period of inactivity? As part of a kiosk system, I have set up a Guest account on a Mac OS X Snow Leopard computer. This account is cleared every time someone logs off the computer. However, plenty of people don't. Is there any way that I can automatically log out this guest account after a period of inactivity?  I tried using the security preferences pane, but that just causes the confirm logout dialog to show up.  I have hacked around the /System/Library/User Templates/ folder, so I can set the screensaver, desktop background, &amp;c. If something like ScriptSaver can solve my problem, I will gladly use it. Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ Solve $u_t=u_{xx}+u_x$ What's this =! operator? Procedure for reducing the degrees of freedom of an arbitrary system What is the meaning of the term "thread-safe"? Does it mean that two threads can't change the underlying data simultaneously? Or does it mean that the given code segment will run with predictable results when multiple threads are executing that code segment? Given the sentence:     Most people who live and work near Washington, DC, would have trouble      imagining dinosaurs walking around the area.   Grammatically, what would ‘imaging’ be called in this case? Present participle or gerund or something?  I mean, if it is a present participle, ‘imagining~’ might be a description of the subject. And if it is a gerund, ‘imagining~’ might be an object of the verb phrase of ‘have trouble’. How do they explain this ‘imagining’ after ‘have trouble’? Where are the Stack Exchange data dumps? $(i)$ $a&gt;0,b&gt;0,p&gt;0,q&gt;0$   $(ii)$ $\frac{1}{p}+\frac{1}{q}=1$     How can we prove it?   ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$ I have the following code:  Using cmd As SqlCommand = Connection.CreateCommand     cmd.CommandText = "UPDATE someTable SET Value = @Value"     cmd.CommandText &amp;= " WHERE Id = @Id"     cmd.Parameters.AddWithValue("@Id", 1234)     cmd.Parameters.AddWithValue("@Value", "myValue")     cmd.ExecuteNonQuery End Using   I wonder if there is any way to get the final SQL statment as a String, which should look like this:  UPDATE someTable SET Value = "myValue" WHERE Id = 1234   If anyone wonders why I would do this:   for logging (failed) statements for having the possibility to copy &amp; paste it to the Enterprise Manager for testing purposes