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Section 13.2 Fundamental Elments

When studying Linear Algebra, we notice two major part,

  1. Solving a system of linear equations.
  2. Finding eigenvalues and eigenvectors.

Subsection 13.2.1 Solving a linear system

Study of Solving systems of linear equations using methods of modern mathematics will require studying of the following topics:

  • Matrices and matrix operations, determinants

  • Vectors

    1. Vector space,
    2. subspace,
    3. linear combination,
    4. spaning set,
    5. linear indepencdence,
    6. basis

  • Three fundamental subspaces namely

    • column spaces,
    • row spces,
    • nullspace
  • Matrices can be factored in many different ways, for this part of the linear algebra, LU and QR factorizations are among the most important ones.

Subsection 13.2.2 Eigenvalues and Eigenvectors

Working with Eigenvalues and Eigenvectors requires studying the following topics.

  • Observing Matrices as linear transformations, relating the trsnsformation to its representation in a given basis, learning about diagonalization.
  • Defining Length of a vector and distance between two vectors,
  • Defining angel between two vectors and intoducing orthognality
  • Defining projections using it to find Least Square solutions.
  • Defining eigenvalues and eigenvectors, diagonalization.
  • Learning Singular Values decomposition and its applications.
  • When learning about these topics other ideas and concepts will develope that are as important as the topics listed here.