-
Recent Posts
Archives
- March 2020
- June 2018
- July 2016
- June 2016
- May 2016
- March 2015
- February 2015
- January 2015
- December 2014
- December 2013
- November 2013
- July 2013
- June 2013
- May 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
Categories
Meta
Pages
Tag Archives: linear algebra
Commutative Algebra 7
Modules Having dipped our toes into algebraic geometry, we are back in commutative algebra. Next we would like to introduce “linear algebra” over a ring A. Most of the proofs should pose no difficulty to the reader so we will … Continue reading
Posted in Advanced Algebra
Tagged ideals, linear algebra, module homomorphism, modules, rings, submodules
Leave a comment
Tensor Product and Linear Algebra
Tensor products can be rather intimidating for first-timers, so we’ll start with the simplest case: that of vector spaces over a field K. Suppose V and W are finite-dimensional vector spaces over K, with bases and respectively. Then the tensor product is the vector … Continue reading
Posted in Notes
Tagged bilinear maps, duals, linear algebra, tensor algebra, tensor products, universal properties
Leave a comment
Elementary Module Theory (IV): Linear Algebra
Throughout this article, a general ring is denoted R while a division ring is denoted D. Dimension of a Vector Space First, let’s consider the dimension of a vector space V over D, denoted dim(V). If W is a subspace of V, we proved earlier that … Continue reading
Posted in Notes
Tagged advanced, algebra, linear algebra, module homomorphism, vector spaces
Leave a comment
Elementary Module Theory (III): Approaching Linear Algebra
The Hom Group Continuing from the previous installation, here’s another way of writing the universal properties for direct sums and products. Let Hom(M, N) be the set of all module homomorphisms M → N; then: (*) for any R-module N. In the case where there’re finitely … Continue reading
Posted in Notes
Tagged cokernels, direct products, direct sums, homomorphism, isomorphism theorems, kernels, linear algebra, modules, submodules, vector space
Leave a comment
Why Do We Need Eigenvalues and Eigenvectors?
[ Prerequisites : basic linear algebra, matrices and determinants. ] Eigenvalues and eigenvectors are often confusing to students the first time they encounter them. This article attempts to demystify the concepts by giving some motivations and applications. It’s okay if … Continue reading
Linear Algebra: Inner Products
[ Background required: basic knowledge of linear algebra, e.g. the previous post. Updated on 6 Dec 2011: added graphs in Application 2, courtesy of wolframalpha.] Those of you who already know inner products may roll your eyes at this point, … Continue reading
Posted in Notes
Tagged advanced, extra, fourier transform, inner product, intermediate, linear algebra, notes, sums
Leave a comment
Matrices and Linear Algebra
Background recommended : coordinate geometry Here I thought I’d give an outline of linear algebra and matrices starting from a more axiomatic viewpoint, instead of merely giving rules of computation – the way it’s usually taught in school. The materials … Continue reading
