Let $V$ be finite-dimensional vector space such that $dim(V)=n$ and consider $T \in L(V)$ and $\lambda_1,...,\lambda_k$ all the different eigenvalues. a) If $W_j$ is the subspace associated to $\lambda_j$ for $j\in{1,...,k}$ prove that $W_1,...,W_k$ are independent  What I know here is that we have to prove that $w_1+...+w_k=0, w_i \in W_i$  If T is diagonalizable, prove that $V=W_1 \oplus...\oplus W_k$  If T is diagonalizable $\exists\beta=${$v_1,...,v_n$} such that $v_i$ is a characteristic vector $\forall i \in${$1,...,n$} how do I prove $V=W_1 \oplus...\oplus W_k$?
I'm trying to show that a finite sum of eigenspaces (with distinct eigenvalues) is a direct sum.   I have $ \alpha : V \to V $. The eigenspaces are $ V_{\lambda_i} = \ker(\alpha - \lambda_i id_V )$ for $ 1 \leq i \leq n $. My attempt at a proof:  $ A + B $ is a direct sum iff $ A \cap B = \{0\} $. If $ v \neq 0 \in V_{\lambda_i} \cap V_{\lambda_j} $ for some $i,j, i \neq j $, then $ \alpha(v) = \lambda_i v $ and $ \alpha(v) = \lambda_j v $. So $(\lambda_i - \lambda_j)v = 0 $, and so $ \lambda_i = \lambda_j $. This is a contradiction, so any pair of the eigenspaces have trivial intersection. Therefore $ \cap_{i=1}^n V_{\lambda_i} = \{0\} $, and so we have a direct sum.  Is this ok?  Thanks
Question Staging SE Been a few blogs on , and I get the point. However, would a staging SE help solve some of these issues like community overlap. I understand , but how can someone not yet a part of a community know to do so. Some of the point of SO is that you often don't know what to ask exactly, and we help clear that up. So, I wonder, rather than &quot;playing hot potato&quot; would it be useful to have a site that's simply for temporarily holding questions and such.  It would be the alternative to migrating to another site where you dump all responsibility on that site. This area, all communities are responsible for. It would be a good place to ask a question if you don't know where to ask. The blog I linked talks about phrasing to a specific audience. For someone that doesn't know, the audience is better at selecting which questions are directed at them. As SE grows, the amount of communities will mean more overlap. Right now it may be reasonable to expect people to be able to predict the best community, but will it always be that way?  Some restrictions:  It would be accessible by all communities. The question could be edited, but not answered. The question could be exposed to more easily tracked/understood/etc. automated janitorial work.  Just a helpful thought. How many times can American Tourist re-enter UK in same 6 month period? Browser back leaves page after going fullscreen in stack snippet Opening a Stack Snippet in full screen should use history.pushState() to add an entry to the browser history stack. Clicking back will then close the snippet rather than leave the page completely. Is there some sort of classification of all minimal non-cyclic groups? Does there exist some sort of classification of all minimal non-cyclic groups (non-cyclic groups, such that all their proper subgroups are cyclic)  I know the following classes of such groups:  1) $C_p × C_p$, where $p$ is a prime  2) $Q_8$  3) $\langle a,b | a^p = b^{q^m} = 1, b^{−1}ab = a^{r}\rangle$, where $p$ and $q$ are distinct primes and $r ≡ 1 \pmod q$, $r^q ≡1 \pmod p$.  (These three classes completely cover the case, when our group is finite: )  4)$C_{p^{\infty}}$, where $p$ is a prime  5)$(\{ \frac{n}{p^m}| m, n \in \mathbb{Z} \}, +)$, where $p$ is a prime  (These two classes completely cover the case, when our group is infinite abelian: )  6)Infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $C_{p}$ for a fixed prime $p$ (Tarski monster groups)  7)Infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $C_{\infty}$ ().  However, I do not know, whether there exists anything, that does not fall into these classes. I only know, that if such groups exist, they have to be infinite non-abelian. What is the correct way to install proprietary ATI Catalyst Video Drivers (fglrx) directly from AMD? I am planning on doing a fresh install of Ubuntu and want to know what is the correct way to install ATI Catalyst Video Driver?      There are multiple valid answers for this question spanning over several versions of Ubuntu. For your convenience an index of each below: Is the usage of text embedded in an image a bad practice, for very controlled uses? What are your thoughts on writing your text in images?  I'm not saying throughout the whole website... But in certain key locations like image carousels. And I'm not talking about the special cases where you have font types that browsers can't render. I'm talking about text that uses standard fonts and is not too big (3/4 bullets).  Should I embed the text in the images that compose the carousel? Or should I apply the text to the images as "real text" HTML?  What are the advantages/ disadvantages of using text in an image? Creating geodatabase feature class using geodatabase table schema with ArcPy? Is there a way in ArcPy to create an empty polygon geodatabase feature class that will acquire the schema of a geodatabase table?  I know that it's possible in ArcCatalog (New -> Feature Class and then import all the fields from a table) but I wonder if it can be written in a Python script.  The CreateFeatureclass_management tool allows only Feature Layers as schema templates. Difference between "I'm home" vs "I'm at home" Logitech LS21 recognized as 2.0 system - Windows 8.1 doesn't detect Subwoofer I got my dad a Logitech LS21 system for Christmas. It's a 2.1 system, but it's recognized in Windows 8 as a 2.0 system (it says there is no subwoofer connected).    How would I go about getting the sub recognized?  It plays some sound now, but it's so quiet you can't hear it without putting your ear right next to it. If $k$ is a positive integer, how to prove that there is no solution for $n$ in positive integers except $n = 1$ for $2^n = nk + 1$ Using article "the" with famous places? I'm editing a children's book about wonders of the World. In this book, the authors are not native speakers(they are XACT BOOK(India)), and they wrote these specific places without article "the":     -It is Taj Mahal.      -It is Tower of Pisa.   At first, I thought it's okay, but there are some pages has article "the":     -It is the Great Pyramid of Giza.      -It is the Temple of Artemis.   These sentences really make me confused. After researching, this website- - gives me some advice:     Do use the before:           names of rivers, oceans, and seas: the Nile, the Pacific          points on the globe: the Equator, the North Pole          geographical areas: the Middle East, the West     deserts, forests, gulfs, and peninsulas: the Sahara, the Persian Gulf, the Black Forest, the Iberian Peninsula      Unfortunately, I'm still not sure that adding "the" before each wonder's name is a right choice. Should I add article "the" or not? I already posted this problem in , but I'm still in vain. Syntax error on print with Python 3 The Many Memes of Meta Catchphrases and concepts that spread from person to person are known as [memes], which, courtesy the Internet, can now explode across the Earth like a highly contagious virus (hence &quot;going viral&quot;). As with their real-life counterparts, some infectious diseases are global (pandemic), while others are endemic to specific regions. Stack Overflow, and now even more predominantly Meta Stack Exchange, have seen more than their fair share of these pathological social constructs spread through the user base. They are now ingrained units of our collective culture as SOpedians (a term which I hate, by the way). Just as travelers' immune systems can be assaulted by new diseases in new places, new users are increasingly likely to be miffed by an ingrained meme and left sitting scratching their heads. I therefore propose that this space be used to document the memes endemic to Stack Exchange's culture. Each meme should be listed in a separate answer and I hope that we as a community will be able to provide greater context to each one. Please actually explain each meme in a way understandable to those not already in the know. Especially considering that a lot of these memes aren't really understandable to newer users who weren't around in the early years of the network, or are based on system features that no longer exist.  On other Stack Exchange sites          [memes]: Implements vs extends: When to use? What's the difference? Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$ Full question: A function $f$ from a space $X, \tau$ to a space $Y, \tau'$ is said to be open if $f(V)$ is open in $Y$ whenever $V$ is an open subset of $X$. Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$.  Attempt: Let $S = \Pi_{i \in I} S_i$ be the product space. Then, let $p_i: S, \tau \to S_i, \tau_i$ be the projection $p_i(s) = s_i$ for each $s \in S$. Then, $p_i$ is continuous for all $i \in I$.  Then, if $V_i$ is open in $S_i$, then $p_i^{-1}(V_i) = S$ is one of the open sets in the basis, namely $\{\Pi_I V_i : V_i \text{ open in } S_i \text{ and } V_i = S_i \text{ for all but at most finitely many i}\}$, and we see clearly that $V_i$ fits the criteria for all $i \in I$ because we defined $V_i$ as being open in $S_i$ and $I$ is a finitely countable set, hence there will be at most finitely many $i$ for which $V_i = S_i$.  Thus, the projection $p_i$ is open for each $i \in I$.  I'm not sure if I tackled this correctly and wanted to make sure I was on the right track or got it right. What are the most accepted ways to visualize the results of an independent two sample t-test? Is a numeric table more often used or some sort of plot? The goal is for a casual observer to look at the figure and immediately see that they are probably from two different populations. How to find the foreach index? What kind of questions can I expect to see on a qualifying exam? Use of Website Category Landing Pages OK, here's the situation: A reasonably simple website with a primary navigation menu including drop-down list of sublevels on hover. Each navigation level may have 3 - 6 sublevels below it, none more important than the other. For example see this mockup:    My question is: What should happen if the user Clicks on one of the primary navigation options (such as 'News') instead of just hovering over and choosing a sublevel?  My instinct tells me that I should allow a click of this navigation option; it's expected user behavior to be able to click navigation options. If clicked then a landing page of some sort will be present listing all the sublevels (latest news, blog posts, case studies'). However there is no content to add to this landing page other than presenting a list of links, which is pretty useless. I could pad it out by adding iconography and descriptions of the sublevels, but that's just content for the sake of it; does 'Latest News' really need a description in order to understand what it is?  If I don't include a landing page then clicking this primary navigation will have to open something but if I open the 'Latest News' page that risks the user not knowing about the other sections of the site.  (The landing page will be useful for breadcrumb navigation too, so I don't really want to do away with it).  Is my thinking that a landing page showing the list of sublevels actually useful even though there is no actual content to display, or is there a better alternative to go with? File upload with Javascript without user intervention