Monday, 21 December 2015

LA2, second attempt on my linear algebra notes

Let us call it LA2. My previous attempt [the 1.5LA] looks horrible 2~3 years after producing it. Hopefully it is better this time. I'm sure that it will as one's understanding on these elementary topics always improve with your understanding on advanced topics, like functional analysis, matrix analysis...etc.

Click here for the notes

Well I compiled this by picking different bits from my revision notes then some careful fixes and some effort to make it smooth. The beginning two-third is a basic linear algebra course, and the last one-third is an advanced but interesting topic to cover: linear programming.

My general principle to arrange the contents is to solve three main problems: to compute the inverse of a matrix, the geometric implication of linear transformations, and the duality principle in linear programming. Some external guidance are also taken, like the classic Linear Algebra from S. Lang.

Multiple references are taken, see p.3 of the text. I've taken [copied] some exercises from those references as well.

Ch.1 Basics: vector and matrix arithmetic
Ch,2 Linear system: reduction, independence, dimension
Ch.3 Geometry I: orthogonality, projection
Ch.4 Determinants: implication in lower dimensions, elementary matrices, matrix inverse, Cramer's rule
Ch.5 Abstract vector space: axioms, function space, inner product space
Ch.6 Linear Transformation: coordinate mapping, change of basis
Ch.7 Eigenvectors: diagonalization, applications
Ch.8 Special operators: Hermitian, unitary operators, orthogonal diagonalization, spectral decomposition
Ch.9 Geometry II: affine space, convex sets, cones
Ch.10 Polytopes and polyhedrons: V-H characterization, Fourier Motzkin elimination
Ch.11 Linear programming: Farkas Lemma, separation, simplex, duality

It was not the main motivation to get the old 1.5LA revamped, but actually I'm writing this for one of my dear friend who is studying computer science who is getting stuck with them, and I hope that there will be more readers other than him :)

And, merry Christmas.

Chris, Dec 2015

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