Let $W$ be a submodule of $V$. Prove that if $V/W$ and $W$ are finitely generated then so is $V$. I have a rough idea of how this works intuitively, $V/W$ and $W$ being finitely generated means that they both have a finitely generating subset, we put $\{a_1, \dots, a_n\}$ to be $W$'s basis, and $\{b_1, \dots, b_m\}$ to be $V/W$'s. Any element $c \in V/W$ can be written as a linear combination of $\{b_1, \dots, b_m\}$ and as $a + b$ such that $a \in W$ and $b \in V$, $a$ is written as a linear combination of $\{a_1, \dots, a_n\}$. $b = c - a$ with $c$ and $a$ being linear combinations of finitely generated elements, this forces $b$ to be finitely generated?
$M$ finitely generated if submodule and quotient are finitely generated. I know this must be easy. I'm new to modules, so perhaps I must be missing  something important.     Let $M$ be an $R$-module. Show that if there exists a submodule $N$ such that $N$ and $M/N$ are finitely generated, then $M$ is finitely generated.   If the exact sequence $0 \longrightarrow N \longrightarrow M \longrightarrow M/N \longrightarrow 0 $ (where the second and third arrows are inclusion and projection respectively) splits then $M$ would be isomorphic to the direct sum of $M$ and $M/N$ and the conclusion would follow (I think). But I can't show that it splits.  Thank you.
Including sections of source code as a listing, while being robust to source code edits I am trying to find out whether $A = \begin{bmatrix}     3 &amp; 1 \\     0 &amp; 2 \\ \end{bmatrix}$ and $B = \begin{bmatrix}     4 &amp; 1 \\     -2 &amp; 1 \\ \end{bmatrix}$ are similar. Both matrices have the same trace, same rank, same determinant and the same eigenvalues. So I think they are similar. The eigenvalues are $3, 2$. Now I'm not sure how to find if they are indeed similar.   I am trying to find if there is a basis for $\mathbb{R^2}$ that contains only eigenvectors but when trying to solve $(3 I - A)v = 0$ there is no solution... What should I do from here? Very basic question: how do I write a short literal in C++?  I know the following:   2 is an int 2U is an unsigned int 2L is a long 2LL is a long long 2.0f is a float 2.0 is a double '\2' is a char.   But how would I write a short literal?  I tried 2S but that gives a compiler warning. Dropping one of the columns when using one-hot encoding My understanding is that in machine learning it can be a problem if your dataset has highly correlated features, as they effectively encode the same information.  Recently someone pointed out that when you do one-hot encoding on a categorical variable you end up with correlated features, so you should drop one of them as a "reference".  For example, encoding gender as two variables, is_male and is_female, produces two features which are perfectly negatively correlated, so they suggested just using one of them, effectively setting the baseline to say male, and then seeing if the is_female column is important in the predictive algorithm.  That made sense to me but I haven't found anything online to suggest this may be the case, so is this wrong or am I missing something?  Possible (unanswered) duplicate: TikZ: remove margin, padding, border around nodes containing images I want to create a simple figure where images are connected with arrows. Here is what I did:  \begin{tikzpicture} \node[](A) {\includegraphics[width=0.2\textwidth]{a}}; \node[right=1mm of A](B) {\includegraphics[width=0.2\textwidth]{b}}; \draw[-&gt;] (A) to (B); \end{tikzpicture}     But I want the arrow to start and end right at the images with no space between. How can I do that? How do I remove duplicate items from an array in Perl? I have an array in Perl:  my @my_array = ("one","two","three","two","three");   How do I remove the duplicates from the array? Having problems with listings and UTF-8. Can it be fixed? On November 11, my American girlfriend flew over to the UK to visit me. She was detained and questioned, and refused entry under the belief she would not leave within those six months. This was due to her having no return flight booked, as we were going to wait until we were together in the UK to book a return flight for either herself, or both of us back to the USA once we decided what exact dates I would be able to travel. Lesson learned. A hard lesson, but we learned!  Anyway, they said although she'll flag up in the future, it won't be a straight ban from entering and she could even try to come back the next day if she so pleased, although it would be unwise if her circumstances had not changed.  Fast forward to December 18. She got herself a job in the States to prove she intends to leave at the end of her stay, and has since applied for a UK Standard Visitor Visa. She posted her documents off to New York earlier today.  Will they send her a letter back refusing to issue her a visa since, as an American citizen, she does not even need one? She wanted to get one to be safe as I told her not to fly to the UK even with return flights booked if she doesn't have a visa, as she risks being denied once again after a long flight. It's hard to get this information on the Internet, and some stories I read said that they were sent a letter back from the UK Embassy denying their visa simply because they don't need one in the first place - even after that person was denied entry.  We're in a bit of a pickle granted, so any help would be greatly appreciated.   Cheers. Regulated AC power supply? unsigned int vs. size_t With ksh I'm using read as a convenient way to separate values: $ echo 1 2 3 4 5 | read a b dump $ echo $b $a  2 1 $  But it fails in Bash: $ echo 1 2 3 4 5 | read a b dump $ echo $b $a     $  I didn't find a reason in the man page why it fails, any idea? How can I send the stdout of one process to multiple processes using (preferably unnamed) pipes in Unix (or Windows)? 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