Glossary for Linear Algebra + Quick Tips

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Thanks Greg! That couldn't have come at a better time for me!
 
Thank you, for this subject
 
And in case you haven't checked it out already, there's also the accompanying http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/CourseHome/index.htm". The video lectures on the page are pretty good, although you may want to download the full ~100 mb file to reduce stuttering if your connection is not up to it.
 
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W. Gilbert Strang's lectures on Linear Algebra are available on-line

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

Lecture 1 - The Geometry of Linear Equations
Lecture 2 - Elimination with Matrices
Lecture 3 - Multiplication and Inverse Matrices
Lecture 4 - Factorization into A = LU
Lecture 5 - Transposes, Permutations, Spaces Rn
Lecture 6 - Column Space and Nullspace
Lecture 7 - Solving Ax = 0: Pivot Variables, Special Solutions
Lecture 8 - Solving Ax = b: Row Reduced Form R
Lecture 9 - Independence, Basis, and Dimension
Lecture 10 - The Four Fundamental Subspaces
Lecture 11 - Matrix Spaces; Rank 1; Small World Graphs
Lecture 12 - Graphs, Networks, Incidence Matrices
Lecture 13 - Quiz 1 Review
Lecture 14 - Orthogonal Vectors and Subspaces
Lecture 15 - Projections onto Subspaces
Lecture 16 - Projection Matrices and Least Squares
Lecture 17 - Orthogonal Matrices and Gram-Schmidt
Lecture 18 - Properties of Determinants
Lecture 19 - Determinant Formulas and Cofactors
Lecture 20 - Cramer's Rule, Inverse Matrix, and Volume
Lecture 21 - Eigenvalues and Eigenvectors
Lecture 22 - Diagonalization and Powers of A
Lecture 23 - Differential Equations and exp(At)
Lecture 24 - Markov Matrices; Fourier Series
Lecture 24b - Quiz 2 Review
Lecture 25 - Symmetric Matrices and Positive Definiteness
Lecture 26 - Symmetric Matrices and Positive Definiteness
Lecture 27 - Positive Definite Matrices and Minima
Lecture 28 - Similar Matrices and Jordan Form
Lecture 29 - Singular Value Decomposition
Lecture 30 - Linear Transformations and Their Matrices
Lecture 31 - Change of Basis; Image Compression
Lecture 32 - Quiz 3 Review
Lecture 33 - Left and Right Inverses; Pseudoinverse
Lecture 34 - Final Course Review

Also on Youtube - starting with Lecture 1 (Spring 2005) -
 
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Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
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Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
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Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
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