Contents

• Contents
• Vectors in Geometry.
• Fields.
• Systems of linear equations.
  • Systems of equations and associated matrices.
  • The Gauss-Jordan algorithm.
  • Solution of a system of linear equations.
    • The general case.
    • A special case: homogeneous equations.
    • The main theorems.
• Introductory examples.
  • An homogeneous differential equation.
  • An homogeneous system of equations.
  • Bank accounts.
• Vector spaces.
  • Vector spaces and vector subspaces.
  • Linear dependence and independence.
  • Generating set of a vector space.
  • Basis of a vector space.
  • Direct sum of two subspaces.
• Linear mappings.
  • Linear mappings.
  • Kernel and Image of a linear mapping.
  • Linear transformations and matrices.
• The algebra of matrices.
  • How to compute the product of two matrices?
  • Matrix inversion.
• More on systems of linear equations.
• Change of basis.
  • Change of basis.
  • Change of basis: general formulae.
  • Change of basis and coordinates.
  • Change of basis and the matrix of a linear mapping.
• Determinants.
  • The determinant of a square matrix.
  • Matrix inversion.
  • Systems of linear equations, once again.
• Eigenvalues and eigenvectors.
  • The characteristic polynomial of a matrix.
  • The theorem of Cayley-Hamilton.
  • An application of Cayley-Hamilton's Theorem.
  • Eigenvalues and Eigenvectors.
  • Similar matrices.
  • Diagonalization of a square matrix.
  • High powers of a square matrix.
  • The exponential of a square matrix.
• Euclidean spaces.
  • Inner product.
  • Verification of the axioms for the ``geometric'' inner product in $\mathbb{R} ^2$.
  • Orthogonality.
  • Orthonormal bases.
• Some geometry: Cauchy-Schwarz inequality.
• Some more geometry.
  • The triangle inequality.
  • The theorem of Pythagoras.
• Various examples of the inequality of Cauchy-Schwarz.

This short tutorial is a companion material to the course on Linear Algebra given at JCT (first year - fall term and spring term). It is intented to help the student, but will replace neither personal lecture notes nor a good textbook. In particular, proofs are generally not given. Sometimes, we give an outline of the proof. References to books are given on the course home page.

Constructive remarks are welcome. Please e-mail to dana@mail.jct.ac.il.