We can prove that diagonalizable matrices with complex values are dense in set of $n \times n$ complex matrices,   I don't think there is many chances but can we show that the set of diagonalizable matrices over $\mathbb R$ with real entries is dense in $M_n(\mathbb R)$?
Suppose that $n \ge 2$. How to prove that the set $\mathcal D \subset M_n(\mathbb R)$ of the diagonalizable real matrices is not dense in $M_n(\mathbb R)$?
-ing vs -in' ending I wonder if the &quot;g&quot; in the -ing forms is pronounced. When I hear it it seems as if it's not pronounced sometimes or just slightly, though sometimes I've been told that I should pronounce &quot;g&quot; for example in &quot;meeting&quot; just to avoid saying &quot;mitten&quot;. So how should I pronounce &quot;-ing&quot;? Sometimes -ing is written in an informal way as -in' such as:  taking takin'  Is the letter &quot;g&quot; in each case pronounced differently? Why aren't people voting for questions? While at least some answers from every question seem to have been up-voted as a "helpful" answer:   about 49% of questions have score of 0 or 1. another 21% of questions have a score of 2.   Most questions seem to have an answer or two with score +5. Is it that we have a lot of fluff questions and not useful ones? Or perhaps it just seems more natural to vote for answers than questions? If we want the most useful info on Stack Overflow to filter to the top, how can this disparity be fixed?  I think this may cause real problems for people getting the badges which require +25 or +100 votes on questions.    I like a lot of the feedback I'm seeing. Two problems are highlighted:   The work flow doesn't promote voting for questions the way it promotes voting for answers. One solution: when a user answers a question, the system should invite them to vote for the question.  If someone cares enough to answer a question, then they care enough to vote for it as well. Users with less than 15 reputation cannot vote for questions or answers. Is this warranted?  Maybe these users could provisionally vote for questions and answers and those ratings would only take effect when the user reaches 15 reputation?   All I ask is this. As you browse Stack Overflow, when you see a good question: vote for it.  If everyone does, then this problem will disappear. I'm trying to compile a document that only works with an up-to-date distribution, so I'd like to know: how do I update my distribution? Please provide screenshots where appropriate.  List of answers           To keep an eye on what’s going on at CTAN you may subscribe to the  or the RSS feed feed://www.mail-archive.com/ctan-ann@dante.de/maillist.xml I need help to solve following problem from Rudin's Mathematical analysis book:    Convergence of the series $\sum a_n$  implies the convergence of $\sum \dfrac{\sqrt {a_n}}{n}$, if $a_n&gt;0$  I tried to construct a suitable convergence sequence $b_n$ such that $\sum b_n$ converges and $a_n \leq b_n$ but, I am not able to find such sequence $b_n$ .  Thanks for the help and sugestions. Let $ f: \mathbb {R} \rightarrow {\mathbb {R}} $ be given by   $$ f (x) = \left \{\begin {matrix} \sin ({\frac {1} {x}}), &amp; \text {if} \phantom {a} x \neq 0; \\  c, &amp; \text {if} \phantom {a} x = 0 \end {matrix} \right. $$ where $ c \in [-1,1]$. For what values ​​of $c$ is there an antiderivative of $f$?  I do not know of a theorem of the form "if and only if" that  it tells me when a function $ f $ has no antiderivative.  Anyone have a suggestion? Are there any versions of t-SNE for streaming data? I am trying to prove $(a)$ is a maximal ideal in $\mathbb{Z}$,  if and only if $a$ is prime number.  Now I wrote: assume $a \in\mathbb{Z}$, while it's not prime number we can write as $a=xy$, for some integers $x$ and $y$. then $(a)\subset (x)$ and $(a)\subset (y)$  while if $(a)$ is maximal ideal, it's not exist $(k)$ such that $(a)\subset (k)\subset\mathbb{Z}$. so $(a)$ cannot be maximal ideal. we can say by contradictory if $a$ is prime, $(a)$ is maximal ideal.  is this correct?  and how to prove the other way. Blank screen after installing nvidia restricted driver Packaging a bike for flying with a bike bag Mirror cells so that a change in either cell updates the other I have 2 cells containing the same picklist values on separate sheets. I want the cells to mirror each other so that whenever I choose a new value in either sheet 1 or sheet 2, the value on the other sheet is updated to match the newly selected value. Is this possible?  UPDATE   Worksheet 1:   Worksheet Name: 'Live Forecast Tool' Cell containing picklist: C3:D3 (merged) Picklist values within range: ='SF MRF'!H5:Y5  Worksheet 2:   Worksheet Name: 'AH' Cell containing picklist: C3:D3 (merged) Picklist values within range: ='SF MRF'!H5:Y5    Cell C3:D3 needs to update to show the value that has been changed on the other sheet. I mean to ask  that $7$ is an arithmetic  expression  but it can also  be  written  as $7x^0$.  which is a constant polynomial  expression.  Every polynomial  expression  is an algebraic  expression  so with  this logic is $7$ an algebraic  expression  or an arithmetic  expression.  Further $7=5+2$, is it an arithmetic  equation  or algebraic  equation  because  it can also be written  in  $7x^0 = 5x^0 + 2x^0$ form. please help.  Thank you. Is there a convenient way to place a figure* at the end of the current page instead of taking up the entire next page in the twocolumn environment? Placement modifiers [Hb] do not work. How to print every Nth character of a string? This line of code does exactly this except I want it to start at the first character not the 7th. $ echo 1234567890abcdefghijklmnopqrstuvwxy | sed 's/.\{1,7\}\(.\{0,1\}\)/\1/g'  What I'm looking for is 19gow Finding net magnetic and electric force on charged particle This is from my textbook, it is not an assigned problem, but I want to understand.   It says:      Consider the situation in the figure, in which there is a uniform electric field in the x direction and a uniform magnetic field in the y direction. For each example of a proton at rest or moving in the x, y, or z direction, what is the direction of the net electric and magnetic force on the proton at this instant?   I believe I need to use the equation   $$ F_{net}=(q\overrightarrow{E})+q(\overrightarrow{v} \times \overrightarrow{B}) $$  But I'm not sure exactly how. We've just started to learn about this, and I want to get a head start. Could anyone put me on the right track? regex error - nothing to repeat In other words, can I do something like  for() {     for {        for {        }     } }   Except N times? In other words, when the method creating the loops is called, it is given some parameter N, and the method would then create N of these loops nested one in another?  Of course, the idea is that there should be an "easy" or "the usual" way of doing it. I already have an idea for a very complicated one. Proving $v(s,p)=2^{p-1}(2s-1)$ is a bijection of natural numbers and $f(s)=2s-1$ is a bijection between natural numbers and odd numbers. How do I prove this function is bijective? $$ v(s,p)=2^{p-1}(2s-1). $$ The domain is natural numbers and the codomain is also the natural numbers.  And this one: $$ f(s)=2s-1. $$ The domain is the natural numbers, and the codomain is the odd numbers in the natural numbers.  With this one I would do this to show it's injective: $$ \begin{align} v(s)&amp;=v(s_1)\\ \implies 2s-1&amp;=2s_1-1\\ \implies (2s_1)/2&amp;=(2s_2)/2\\ \implies s=s_1 \end{align} $$ So it's injective since if $v(s)=v(s_1)$ then $s=s_1$.  And to show it's surjective $f(s)=y$: $$ \begin{align} y&amp;=2s-1\\ \implies s&amp;=(y+1)/2 \end{align} $$ Then the function must be surjective since every $y$ is the same as the codomain for $f$.  Am I correct? How to find windows 10 activation Key Where can it be find?  At activation panel i can only find option to change the activation key. The current key is not presented on this screen. I need a step-by-step, simple and easy way to configure swappiness. After using `networksetup -setdnsservers Wi-Fi "8.8.8.8" "8.8.4.4"` how do I clear them? I'm looking to script a temporary DNS server change to override the DHCP response I get from the network. I've figured out I can use networksetup but once we're done testing. Before I push to production, I figure I should know how to undo this.  How can I script to clear these changes and restore the settings the network provides now as if I never ran the following script?  networksetup -setdnsservers Wi-Fi "8.8.8.8" "8.8.4.4"   It's not feasible to save any the previous settings and I'm looking to remove the setting so that the actual new values can be retrieved from the network as opposed to saving what the settings were previously so I don't save an old value that is no longer correct.  Basically, I'm looking to ipconfig /renew on macOS.