Linear Algebra & Geometry Syllabus

• Vector in 2-space and in 3-space and their operations.
  • Dot product, cross product and box product.
  • Lines and planes in 3-space.
  • Orthogonal projections.
• Matrices and their operations.
  • Strongly reduced matrices.
  • Matrix form of linear systems of equations and their solutions with geometrical applications.
  • Matrix equations and inverse of a matrix. Determinants.
• Vector spaces:
  • definition, examples and applications.
  • Sub-vector spaces and main operations with them.
• Linear combination and linearly dependent vectors.
  • How to extract linearly independent vectors from a set.
  • Bases of a vector space.
  • Dimension of a vector space. Dimension of finitely generated subspace.
• Space of polynomials. Grassmann’s relation.
  • Linear maps.
  • Image of a linear map.
  • Injective and surjective linear maps.
  • Isomorphisms.
• Matrix of a linear map.
  • Endomorphism and square matrices.
• Eigenvalues and eigenvectors.
  • Eigenspaces of matrix endomorphisms.
  • Characteristic polynomial of an endomorphism.
  • Diagonalization of and endomorphism.
• Orthonormal bases, orthonormal matrices.
  • Gram-Schmidt’s algorithm.
  • Diagonalization of real symmetric matrices using orthogonal matrices.
  • Quadratic forms and the sign that they can take in a point.
• Metric problems:
  • distance between two points,
  • two lines,
  • and a point and a line.
• Quadratic geometry:
  • conic curves, and spheres.
  • Non-degenerate quadrics in canonical form.
  • Recognising a quadric surface.

• Machine arithmetic, machine numbers, rounding error. Conditioning of a numerical problem, stability of an algorithm.

• Approximation of functions and data :
  • polynomial interpolation and piecewise polynomial interpolation (spline).
  • Main results about convergence.
• Linear systems:
  • conditioning and numerical direct methods.
  • Matrix factorizations PA=LU, Choleski, QR and their main applications.
• Eigenvalues of matrices:
  • conditioning and numerical methods (powers, inverse power, QR (basics notions)).
  • Singular values decomposition of matrices and its main applications.