Applied Linear Algebra and Matrix Analysis

• Section 0 Introduction: Introduction
• Section 1 group, ring, and fields. (This is a generalization of real number field R.) Math5110Sec1 (extra: FieldsClassWork)
• Section 2 matrix operations. Math5110Sec2 (extra: MatrixClassWork)
• Section 3 general vector spaces over a field. (This is a generalization of subspaces of R^n) Math5110Sec3
• Section 4 Independence and basis (of any vector spaces) Math5110Sec4 (extra: ClassWorkBasis)
• Section 5 coordinates, matrix of linear transformations and change of coordinate Math5110Sec5
• Section 6 determinants Math5110Sec6 (extra: ClassWorkDet)
• Section 7 diagonalization and eigenspaces. Math5110Sec7
• Section 8 Jordan Canonical forms, Cayley-Hamilton theorem, minimal polynomials. Math5110Sec8
• Section 9 Perron-Frobenius Theorem, Dynamical system, Markov chains, etc. Math5110Sec9
• Section 10 inner product spaces (this is a generalization on dot products.) Math5110Sec10 (Extra: InnerProduct,  Sec InnerProduct)
• Section 11 General Least squares problems, data fitting, function approximation. Math5110Sec11
• Section 12 Fast Fourier Transform. Math5110SecFFT
• Section 13 Symmetric matrices and quadratic forms  Math5110Sec13
• Section 14 Singular value decompositions(SVD) Math5110Sec14
• Section 15 Principal component analysis(PCA). Math5110Sec15
• Extra Topics: (Hilbertspaces_Multi-linear spaces_More , Grassmannian and distances between spaces)
  • Matrix Calculus.
  • Hilbert Spaces.
  • Convergence of sequences and series in a normed vector space.
  • Linear Programming
  • An Introduction to algebraic and spectral graph theory
  • Haar Bases and Haar wavelets
  • Hadamard matrices.
  • The group of unit quaternions, SU(2), and the representation of rotations in SO(3) by unit quaternions
  • The geometry of the orthogonal groups O(n) and SO(n), and of the unitary groups U(n) and SU(n).
  • Exterior algebra and tensor algebra
  • Multilinear Algebra
  • Affine space and affine maps
  • Duality norm and dual norms

 

• References:

1.  Finite-dimensional linear algebra, Mark S. Gockenbach, CRC Press.
2.  Introduction to Linear Algebra, Gilbert Strang, Wellesley-Cambridge Press
3.  Applied Linear Algebra and Matrix Analysis, Shores, Thomas S., Springer
4.  Applied Linear Algebra, Olver, Peter J., Shakiban, Chehrzad, Springer
5.  A Second Course in Linear Algebra, S. R. Garcia, R. A. Horn, Cambridge University Press
6.  Matrix Analysis and Applied Linear Algebra, C. D. Meyer, SIAM, 2000.
7. Advanced Linear algebra, Steven Roman, GTM, Springer 3rd edition
8. Introduction to Applied Linear Algebra Vectors, Matrices, and Least Squares https://web.stanford.edu/~boyd/vmls/vmls.pdf
9. Linear Algebra and Optimization with Applications to Machine Learning: Volume I: Linear Algebra for Computer Vision, Robotics, and Machine Learning and Volume II: Fundamentals of Optimization Theory with Applications to Machine Learning by Jean Gallier and Jocelyn Quaintance
10. Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning https://www.cis.upenn.edu/~jean/gbooks/geomath.html