Let $V$ be a vector space. In order to show that any linearly independent set in $V$ can be extended to form a basis, or equivalently, any basis of a subspace of $V$ can be extended to a basis of $V$, is it required to use choice principles (i.e. Axiom of Choice or weaker forms of Choice). If so, what is the weakest choice principle that can prove this statement?
In discussing , I noted that while the statement:     Any vector space has a basis   is equivalent to the axiom of choice, I wondered if the statement that:     Any vector space either has a finite basis or an infinite set of linear independent vectors   was weaker than the axiom of choice. It feels weaker - it feels like you can inductively define a countably infinite set of linearly independent vectors without full choice, but I'm not sure if that is possible with ZF alone, or requires all of the Axiom of Choice, or is weaker but requires some form of choice.
I am new to this and not sure what to make of all the options available. I am interested in putting a machine-learning-related preprint, which will potentially be submitted to places like NIPS/ICLR/ICML in the future.   arXiv.org perpetual, non-exclusive license to distribute this article (Minimal rights required by arXiv.org)  Creative Commons Attribution license (CC BY 4.0)  Creative Commons Attribution-ShareAlike license (CC BY-SA 4.0)  Creative Commons Attribution-Noncommercial-ShareAlike license (CC BY-NC-SA 4.0)  Creative Commons Public Domain Declaration (CC0 1.0) How to redirect users to custom lostpassword page? How can I ensure my Samsung Series 7 is actually using the Radeon switchable graphics? I have a Samsung Series 7 with the Radeon 6750M switchable graphics. The ATI software lets you force programs to use the dedicated card. However, I'm not convinced it is actually ever using it, as the frame rates on some non-taxing games (Portal, TF2) are merely OK. To make matters worse, it looks like the ATI Catalyst Control has vanished from my laptop. To make it even worse, you can't download the driver and ATI CCC from ATI's site, you need to download it from Samsung, and the ZIP they provide is corrupted.   How can I ensure my Samsung Series 7 is actually using the Radeon switchable graphics? Real valued analytic function defined on a connected set is constant Eulerian Paths in a Complete Graph $K_5$ As the title suggests, I’m simply looking for a method to compute the number of Eulerian Paths in the complete graph $K_5$, where all paths start from the same node. Since there are 5 nodes, all Eulerian paths must be Eulerian circuits as well.  Thanks. Homology of $S^1 \times (S^1 \vee S^1)$ I'm trying to solve question 2.2.9(b) in Hatcher's Algebraic Topology.   Question: Calculate the homology groups of $X=S^1 \times (S^1 \vee S^1)$.  My attempt: I try to use the Mayer-Vietoris sequence.   Let $A=S^1 \times (S^1 \vee \text{small bit}) =S^1\times S^1$ and $B=S^1 \times ( \text{small bit} \vee S^1)=S^1 \times S^1$. (Unsure how to express this, but hopefully this is clear enough.)   Then $A\cap B=S^1\times \{\text{point}\}=S^1$ and $A\cup B=X$.  The Mayer-Vietoris sequence then gives us an exact sequence in reduced homology as follows:  $$0 \to \mathbb{Z}^2 \to \tilde{H}_2(X) \to \mathbb{Z} \to \mathbb{Z}^2 \to \tilde{H}_1(X) \to 0.$$  Is it possible from this exact sequence to determine the homology groups, in some algebraic fashion? Or do I need to determine what the maps are? How do I determine the maps? Joining non-spatial CSV file to spatial data (shapefile) using QGIS? I have a CSV file which has postcode data in a field, but it is non-spatial e.g. no Easting and Northing.   I have a shapefile with postcode in a field and some additional data.  I want to join the CSV with the shapefile based on attributes (postcode fields). I know I can do this in ArcMap, but how can I achieve the same thing in QGIS? I would like to convert a large LaTeX project (i.e. spanning multiple files) into plain text. The purpose is generation of statistics, so representing mathematics is not an issue. In fact, all mathematics is ideally ignored.  I have found  but could not get it to run. It seems unfinished, but is exactly what I am looking for otherwise. Using apt-get to remove packages and dependencies Convert a string representation of a hex dump to a byte array using Java? I am looking for a way to convert a long string (from a dump), that represents hex values into a byte array.  I couldn't have phrased it better than the person that posted .  But to keep it original, I'll phrase it my own way: suppose I have a string "00A0BF" that I would like interpreted as the  byte[] {0x00,0xA0,0xBf}   what should I do?  I am a Java novice and ended up using BigInteger and watching out for leading hex zeros. But I think it is ugly and I am sure I am missing something simple. How do I choose grid and block dimensions for CUDA kernels? How to add sysadmin to user in SQL Server 2008 when no sysadmin accounts exist I have a SQL server 2008 instance running. Unfortunately during testing I deselected the sysadmin rights for my login and now can't readd it (because I don't have sysadmin rights).  There are no other sysadmin accounts for the instance except SA.  I was set up for just Windows Authentication so I hacked the LoginMode in the registry to 2 so I could login as SA using Sql Authentication. This does indeed set the login mode to Mixed, however the SA user is disabled by default and I can't reenable it because I am don't have sysadmin rights.  How do I enable the SA login so I can get in and reassign the sysadmin to my normal account? Is there a registry setting for this too or is it stored in the master database? Include sorting of accepted answer on profile I have few things on my mind:   I viewed my profile and I wanted to sort my answers by  "Accepted Answer, Votes" Wanted to know how many answers I've made that is marked as  the accepted answer. Sort by Most liked Question: I'm referring to my question who  have the most numbers of favorites. I wanted to sort the questions I created order by accepted or  unaccepted questions.    1 &amp; 2 seems useful to me but 3 &amp; 4 I don't think are that useful. But who knows, someone in this site might find it useful one day!  My POV: My answer is only sorted by Votes, I also want to see a list of accepted  answers on my profile probably not just to know how many accepted answers  I have but also to know what are the accepted answers that I have. This question relates to a discussion on . Euclid's proof of the infinitude of primes is an indirect proof (a.k.a. proof by contradiction, reductio ad absurdum, modus tollens). My understanding is that Intuitionists reject such proofs because they rely on the Law of the Excluded Middle, which they don't accept. Does there exist a direct and constructive proof of the infinitude of primes? What are complex numbers, actually? Every time I read the tips for installing an SDK, IDE, some extension and so on, it says, that I should unpack them in /opt folder. Why do I need to do that?   When I was installing Ubuntu, I read that I should set only 10-20 GiB for the / file system and remaining space set for /home. So should I extend the space for root folder, or leave all stuff at /home? Is there any difference? Given an alphabet of size $n$, how many strings of length $c$ contain every single letter of the alphabet at least once?  I first attempted to use a recurrence relation to work it out:  $$ T(c) = \left\{ \begin{array}{cr} 0 &amp;\mbox{ if $c&lt;n$} \\ n! &amp;\mbox{ if $c = n$} \\ T(c-1) \cdot n \cdot c &amp;\mbox{ if $c &gt; n$} \end{array} \right. $$  As there's no strings that contain every letter if c &lt; n, and if c = n then it's just all permutations. When c > n you can take any string of size (c-1) that contains all letters (of which there are $T(c-1)$ to choose from), you choose which letter to add (of which there are $n$ choices) and there are $c$ different positions to put it. However, this gives out results that are larger than $n^c$ (the total number of strings), so it can't be right, and I realised it was because you could count some strings multiple times, as you can make them taking different inserting steps.   Then I thought about being simpler: you choose n positions in the string, put each letter of the alphabet in one of those positions, then let the rest of the string be anything:  $$ {c\choose{n}} \cdot n! \cdot n^{c-n} $$  But again this counts strings multiple times.  I've also considered using multinomial coefficients, but as we don't know how many times each letter appears in the string it seems unlikely they would be much help. I've also tried several other methods, some complicated and some simple, but none of them seem to work.  How would you go about working out a formula for this? I'm sure there's something simple that I'm missing. I have a following problem: when I plug-in my USB flash drive, it isn't shown in Windows Explorer or recognized by system at all. In device properties, it is shown as:     USB MEMORY BAR     There's no partitions on the drive, so I can't format it:    DISKPART:     Virtual Disk Service error:   There is no media in the device.   HP USB format tool:     There is no media in the specified device.   Any help will be greatly appreciated! Thank You! Mathematical expression to form a vector from diagonal elements I would like to know is there any way to express mathematically (using matrices multiplication, addition, etc.) a vector which is formed from the diagonal elements of a matrix. For example, I have a matrix called $\mathbf{M}$ and I want to create a vector  $\mathbf{v}$ such that $\mathbf{v}= \text{diagonal elements of } \mathbf{M} $  Any matrix algebra or operation do that ? Thank you. We now that Fermat's last theorem is true so there are not positive integer solutions to $$x^n+y^n=z^n$$ for $n\in\mathbb{N}$ and $n&gt;2$.  But what about if $n\in\mathbb{R}$ or $n\in\mathbb{R}^+$?