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Welcome to ProofWiki, the online compendium of mathematical proofs. Log in or create an account to contribute! Welcome to ProofWiki! Welcome to ProofWiki.org! ProofWiki is an online compendium of mathematical proofs! Our goal is the collection, collaboration and classification of mathematical proofs. If you are interested in helping create an online resource for math proofs feel free to Create an account and contribute! Thanks and enjoy! If you have any questions, comments, or suggestions please contact webmaster at proofwiki.org , post on the discussion page, or contact one of the ProofWiki sysops. To see what's currently happening in the ProofWiki community, visit the community portal. Proof Index ~ Definitions ~ Sandbox Site Statistics Proofs:5,401 Definitions:4,091 Users:1,045 Quick Tips How to start a new page/proof. Answer or post a question about page editing. Check the editing help page for help with $\LaTeX$ and wiki editing. If so, read the house style rules. Check the recent changes. Want to help ProofWiki? Find out how here. Latest Proof: Ring of Integers is Principal Ideal Domain/Proof 2 on 17 May 2012 by Prime.mover Top 10 Wanted Proofs Frobenius's Theorem --Matt Westwood 18:38, 8 November 2008 (UTC) Wedderburn's Theorem --Matt Westwood 18:38, 8 November 2008 (UTC) Whitney Immersion Theorem --Matt Westwood 08:09, 18 January 2009 (UTC) Whitney Embedding Theorem --Matt Westwood 08:09, 18 January 2009 (UTC) Topological h-Cobordism Theorem --Matt Westwood 08:09, 18 January 2009 (UTC) Thurston's Geometrization Conjecture --Matt Westwood 08:09, 18 January 2009 (UTC) Tartaglia's Formula --Matt Westwood 06:39, 15 March 2009 (UTC) Burnside's Theorem --Joe (talk) 16:50, 16 March 2009 (UTC) Abel-Ruffini Theorem [1] --Matt Westwood 20:45, 16 March 2009 (UTC) Central Limit Theorem --HrMeyer 20:53, 23 April 2009 (UTC) Want to do something different? Check here for articles linked to but not created, or finish a stub article. News April 1, 2012 Now the Riemann Hypothesis has finally been solved (who would have guessed that it would be soluble relying solely upon analysis of elementary functions?) someone's going to have to go and post it up. I'll get there but I'm bogged down in tidying up the Relation Theory category. --prime mover March 11, 2012 Migrating to a different RSS renderer since the one we currently use seems to no longer be maintained. It appears an upstream version in 1.19 is being supported, going to switch to that. --Joe (talk) February 12, 2012 Trying out the new 2.0 beta of MathJax. --Joe November 30, 2011 Check out the forthcoming AI Mashup Challenge - fancy a mathematics conference in sunny Greece? November 10, 2011 Still getting a lot of what I think are spam accounts being created (since the email confirmation messages all bounce back). Trying to combat this by checking all IP's against SORBS. --Joe (talk) 11:50, 10 November 2011 (CST) November 9, 2011 So, spam sucks! We're moving back to a user based, email authenticated edit system ... effective immediately. If anyone has doesn't like this, let me know. --Joe (talk) 05:49, 9 November 2011 (CST) November 7, 2011 600 registered users. Less than 2 months for 100 more users to join, but quite a few of those were spamming accounts and have now been blocked. --prime mover 17:11, 7 November 2011 (CST) September 30, 2011 Using ReCaptcha for account creation and anonymous posts. Hopefully this will cut down on some of the spam we've been seeing recently. --Joe (talk) 16:36, 30 September 2011 (CDT) September 12, 2011 500 registered users. Just over 3 months for 100 more users to join. --prime mover 00:27, 12 September 2011 (CDT) September 3, 2011 The 4000th proof has been completed. It's a big one, the long-awaited Gelfond-Schneider Theorem. --prime mover 04:25, 3 September 2011 (CDT) September 1, 2011 The 4000th proof page has been added, although the proof itself still needs to be done. Show All News Proof of the Week Linear Transformation as Matrix Product Theorem Let $T: \R^n \to \R^m, \mathbf x \mapsto T\left({\mathbf x}\right)$ be a linear transformation. Then $T\left({\mathbf x}\right) = \mathbf A_T \mathbf x$, where $\mathbf A_T$ is the $m \times n$ matrix defined as: \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \mathbf A_T\) \(=\) \(\displaystyle \begin{bmatrix} \\ \\ \\ T\left({\mathbf e_1 }\right) & T\left({\mathbf e_2 }\right) & \cdots & T\left({\mathbf e_n }\right) \\ \\ \\ \end{bmatrix}\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)                     where $\left \{ {\mathbf e_1, \mathbf e_2, \cdots, \mathbf e_n} \right\}$ is the standard basis of $\R^n$. Proof Let $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$. Let $\mathbf I_n$ be the identity matrix of order $n$. \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \mathbf x_{n \times 1}\) \(=\) \(\displaystyle \mathbf I_n \mathbf x_{n \times 1}\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)          definition of left identity           \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)          Identity Matrix is Identity:Lemma           \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} \\ \\ \\ \mathbf e_1 & \mathbf e_2 & \cdots & \mathbf e_n \\ \\ \\ \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)          definition of standard basis           \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \sum_{i=1}^n \mathbf e_i x_i\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)          definition of matrix multiplication           \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle T\left({\mathbf x}\right)\) \(=\) \(\displaystyle T\left({\sum_{i=1}^n \mathbf e_i x_i}\right)\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)                     \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \sum_{i=1}^n T\left({\mathbf e_i}\right)x_i\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)          definition of linear transformation           \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} \\ \\ \\ T\left({\mathbf e_1}\right) & T\left({\mathbf e_2}\right) & \cdots & T\left({\mathbf e_n}\right) \\ \\ \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \)          definition of matrix multiplication           That $\mathbf A_T$ is $m \times n$ follows from each $T\left({\mathbf e_i}\right)$ being a member of $\R^m$ and thus having $m$ rows. $\blacksquare$ Also see Matrix Product as Linear Transformation Sources For a video presentation of the contents of this page, visit the Khan Academy.