Why if $A \in \textsf{M}_{n\times n}(F)$ and  $$I \notin \operatorname{span} \{A,A^2,\dots,A^n\}$$  so $\det (A)=0$?  I understand that $A$ is not invertible if $I \notin \operatorname{span} \{A,A^2,\dots,A^n\}$, but can't explain why.
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.   I was thinking of showing $A$ is not invertible, meaning it has an eigenvalue of $0$. Since no matter the power you give to $A$, it is still not the identity, you can deduce that at least one eigenvalue is indeed $0$. However this doesn't work if you have different eigenvalues in the Matrix, so I got stuck.  Help will be most appreciated.
Reputation items in the achievements dropdown don't add up I'm trying to figure out a good workflow for developing scripts and add-ons in Blender. Here's what I'm currently doing as I try my luck with a PyNode add-on:   Open blender Create a text editor panel Open my script (external text file) Edit my script in a nice external editor Reload the script Save the .blend file Run the script   Now that I can see what my code does, this Blender instance has been modified, so I can't (make changes to my script and) run my script again. So I do this:   Close blender Open blender with the .blend from step 6 and repeat steps 4-7, over and over and over..   Is there a better workflow for this sort of thing? Can not move polygons but only vertices of an object by bmesh! Multiplicative order and powers Let's consider two number $a$ and $b$ such as $gcd(a,b)=1$. Can you explain me intuitively why there exist $n&gt;0$ such as $a^{n} \equiv 1 \space (mod \space b)$  ? I've been informed by those who know much more about Harry Potter than I that Hogwarts is protected with a Disillusionment Charm so all muggles will see is ruins with a "Danger: Keep out!" sign around them.  (Apparently this was in one of the first three books, but I'm not sure where it was.)  My first thought was, "Well, a danger sign would encourage some muggles to investigate!"  Which leads to what I think is an obvious question: Will a disillusionment charm on a place like that still work when a person is real close to the location?  Will it work from inside the location?  For example, if a curious muggle made it to the gates of Hogwarts and started walking around inside, would they still see it as abandon and in ruins, or once they entered, would they see it as it is? In my opinion the accepted answer should permanently stick as only the original question submitter has the ability to accept. How can I install a package without root access? I have no root access on this machine.  I would like to know if there is a way I can download Ubuntu packages and install them as non-root?  Probably in my ~/bin or ~/usr/share or something like that? Would that work? NSString tokenize in Objective-C What is the best way to tokenize/split a NSString in Objective-C? I was looking for alternate proofs of the theorem that "a vector space $V$ of dimension greater than $1$ over an infinite field $\mathbf{F}$ is not a union of fewer than $|\mathbf{F}|$ proper subspaces" and possible generalizations.  A simple measure-theoretic proof over $\mathbb{R}$ is as follows: By countable additivity the sum of the measures of any collection of subspaces is zero since the measure of each subspace is zero, which is a contradiction.  I would like to look at proofs over arbitrary infinite fields and would like to know if similar statements hold for say modules (finitely generated or otherwise) over infinite rings. When is it appropriate to end a sentence in a preposition? Like many others, I commonly find myself ending a sentence with a preposition.  Yes, it makes me cringe.  I usually rewrite the sentence, but sometimes (in emails) I just live with it.  To, with... you know who you are.  Should I keep fighting myself on this one, or is it okay in some circumstances? Is there a printable version of a reference manual for all TeX primitives? Passing functions to setTimeout in a loop: always the last value? Are two norms equivalent if they induce the same topology on a vector space? Whenever two norms are equivalent in the sense that $\|x\|_1\le c_1\cdot \|x\|_2$ and $\|x\|_2\le c_2\cdot \|x\|_1$, they generate the same topology. Is the reverse also true, i.e. if a topology is generated by two different norms, are the norms equivalent in the above sense? We know this to be true for $\mathbb{R}^n$, but is it generally true, and if not, what are some counterexamples?  What about a Hausdorff, translationally invariant vector space topology generated by two different metrics, are the metrics equivalent in the sense $d_1(x,y)\le c_1\cdot d_2(x,y)$ and vice versa? Could humans choose to establish fibre digesting colonies in our guts? 2 different bibliographies I'm writing my thesis and I would like to make 2 "bibliography" references: The first one would be for the publications I'm in and the second one for external references. I've made one *.bib file right now and it's working but I have no idea how I should work with 2 *.bib files.  Any suggestion will be appreciated. Do I have to peel red potatoes before baking them? I recently made some potato wedges in the oven. I shared this recipe with my relative, who admonished me for not properly peeling the red potatoes before baking them.  She claims that there are disease and viruses that lie on the skin, so peeling it will get rid of it.  I bake it, so I assume that even the high heat in the oven would kill bacteria and germs. And besides that, I make sure to wash potatoes thoroughly before cooking them.  Am I in the wrong here? Should I have properly peeled the potato wedges before baking them? Personally I feel they add more flavor, and from all the pictures on the recipes online, it seems they also do not peel it. Why does this sort ignore the +/- character prefix? TikZ externalization with absolute path and MiKTeX distribution This code:  \documentclass[11pt,openright]{book} \usepackage[T1]{fontenc} \usepackage[latin9]{inputenc} \usepackage[british]{babel}  \usepackage{tikz} \usetikzlibrary{external} \tikzexternalize[prefix={"C:/Users/MyName MySurname/Documents/LaTeX/Figures/"}]  %\tikzexternalize[prefix={Figures/}]   \begin{document} \begin{tikzpicture}     \draw (0,0) circle (1cm); \end{tikzpicture} \end{document}   gives me the error:     ! I can't write on file `"C:/Users/MyName   MySurname/Documents/LaTeX/Figures/MyMWE-figure0.md5"'.   I'm working with Windows 10, MiKTeX 2.9 distibution.  Following the advice Tarass gave me on a comment of this question:  and as indicated here:  I've registered a user-managed TEXMF directory but I still get the same error.  With \tikzexternalize[prefix={Figures/}] all works perfectly but, since I'm using LyX, the files are stored in a LyX temp dir, hence I loose them when I close the program (making externalization useless), so I need to use an absolute path. Why not use "which"? What to use then? When looking for the path to an executable or checking what would happen if you enter a command name in a Unix shell, there's a plethora of different utilities (which, type, command, whence, where, whereis, whatis, hash, etc).  We often hear that which should be avoided. Why? What should we use instead? The only things I've found so far are a cart method (which ceased working in 1.6), a dispenser method (utilizing the decay on dropped items, but this requires occasional feeding), and a binary counter (which won't work for a single input but rather a fluctuating one).  I want to have a circuit that when I apply an input signal, the output signal doesn't change for 2 to 5 minutes.  When the input turns off, I again want a 2 to 5 minute delay.  I could accomplish this with (roughly) 240 to 600 redstone repeaters, but this is far in excess of being useful.  The only method I can think of, but would require a XOR gate (on the button-input and final output) to start and stop a clock attached to a binary counter.  While I can make room for a device of this size, I was hoping for something simpler.