Linear Algebra Problems? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:08:12Z http://mathoverflow.net/feeds/question/15050 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15050/linear-algebra-problems Linear Algebra Problems? zeb 2010-02-11T23:30:30Z 2012-03-03T22:16:29Z <p>Is there any good reference for difficult problems in linear algebra? Because I keep running into easily stated linear algebra problems that I feel I <em>should</em> be able to solve, but don't see any obvious approach to get started.</p> <p>Here's an example of the type of problem I am thinking of: Let $A, B$ be $n\times n$ matrices, set $C = AB-BA$, prove that if $AC=CA$ then $C$ is nilpotent. (I saw this one posed on the KGS Go Server)</p> <p>Ideally, such a reference would also contain challenging problems (and techniques to solve them) about orthogonal matrices, unitary matrices, positive definiteness... hopefully, all harder than the one I wrote above.</p> http://mathoverflow.net/questions/15050/linear-algebra-problems/15051#15051 Answer by Jonas Meyer for Linear Algebra Problems? Jonas Meyer 2010-02-11T23:34:44Z 2010-02-11T23:48:05Z <p><a href="http://books.google.com/books?id=DQP3AIlrCP0C&amp;lpg=PP1&amp;client=firefox-a&amp;pg=PP1#v=onepage&amp;q=&amp;f=false" rel="nofollow">Halmos's Linear Algebra Problem Book</a>. It contains problems, then hints, then solutions. There is a variety of difficulty levels, and some of the problems are very easy, but some are challenging. The book is designed to be a supplement for learning linear algebra by problem solving, so it may not have the focus you're looking for.</p> http://mathoverflow.net/questions/15050/linear-algebra-problems/15053#15053 Answer by Wanderer for Linear Algebra Problems? Wanderer 2010-02-11T23:38:07Z 2010-02-11T23:38:07Z <p>Indeed, Halmos is a very good reference. You will also find some nice problems in <a href="http://www.amazon.com/Berkeley-Problems-Mathematics-Paulo-Souza/dp/0387008926" rel="nofollow">Berkeley problems in mathematics</a> and on the website of the <a href="http://www.imc-math.org/" rel="nofollow">International Mathematics Competition</a></p> http://mathoverflow.net/questions/15050/linear-algebra-problems/15054#15054 Answer by Mariano Suárez-Alvarez for Linear Algebra Problems? Mariano Suárez-Alvarez 2010-02-11T23:40:01Z 2010-02-11T23:40:01Z <p>Google will find for you V. Prasolov's <em>Problems and Theorems in Linear Algebra</em>, which has beautiful more or less hard problems.</p> http://mathoverflow.net/questions/15050/linear-algebra-problems/15120#15120 Answer by Portland for Linear Algebra Problems? Portland 2010-02-12T15:30:10Z 2010-02-12T15:30:10Z <p>In addition to those mentioned above, there is <em>Linear Algebra: Challenging Problems for Students</em> by Fuzhen Zhang </p> http://mathoverflow.net/questions/15050/linear-algebra-problems/15130#15130 Answer by Alekk for Linear Algebra Problems? Alekk 2010-02-12T16:49:27Z 2010-02-12T16:49:27Z <p>you could also browse the linear algebra section of <a href="http://www.mathlinks.ro/index.php?f=349" rel="nofollow">AoPS</a>.</p> http://mathoverflow.net/questions/15050/linear-algebra-problems/17042#17042 Answer by Sunni for Linear Algebra Problems? Sunni 2010-03-04T02:09:03Z 2010-03-04T02:09:03Z <p>Let me introduce you a good reference, IMAGE. At the end of IMAGE there is a section called IMAGE Problem Corner: Solutions of Old Problems and New Problems. You may enjoy solving these problems and read solutions by others. See <a href="http://www.math.technion.ac.il/iic/IMAGE/" rel="nofollow">http://www.math.technion.ac.il/iic/IMAGE/</a></p> http://mathoverflow.net/questions/15050/linear-algebra-problems/90144#90144 Answer by Gerhard Paseman for Linear Algebra Problems? Gerhard Paseman 2012-03-03T20:10:41Z 2012-03-03T20:10:41Z <p>I will take this opportunity to post my favorite linear algebra problem. I call it 0 not equal to 1.</p> <p>Let A be an nxn 0-1 matrix with nonzero determinant. Show that there is a 1 in every row and in every column of A, and further there is a permutation matrix P so that PA has a diagonal of all 1's.</p> <p>Let B be an nxn 0-1 matrix with nonzero determinant. We cannot show that there is a 0 in every row and in every column, so assume B also has this property. Are there nxn permutation matrices P and Q such that PBQ has all 0's on the diagonal? If not, how small a trace can one guarantee?</p> <p>Gerhard "Ask Me About System Design" Paseman, 2012.03.03 </p> http://mathoverflow.net/questions/15050/linear-algebra-problems/90155#90155 Answer by Federico Poloni for Linear Algebra Problems? Federico Poloni 2012-03-03T22:16:29Z 2012-03-03T22:16:29Z <p>If you happen to know a little bit of Italian, another good resource is <em>Problemi risolti di algebra lineare</em>, by Broglia, Fortuna, Luminati.</p> <p>(By the way, if you have never done it, reading a math book in another language is often easier than it seems at first sight)</p>