Course details

Linear Algebra and Geometry

ILA Acad. year 2004/2005 Winter semester 4 credits

Current academic year

Matrices and linear systems. Matrix operatins. Rank of a matrix. The homogenious and non-homogenious linear system and its solution. The inverse matrix. The determinant. The cofactor expansion. The adjoint matrix. Cramer's rule. Vector spaces. The basis and the dimension. The transition matrix. The inner product. Orthogonalization. The orthogonal projection. Operations with vector spaces. The eigenvalues and eigenvectors. The quadratic forms, conic sections and quadratic surfaces. The linear analytic geometry. The vector calculus in R^3.

Guarantor

Krupková Vlasta, RNDr., CSc.

Language of instruction

Czech

Completion

Examination

Time span

  • 26 hrs lectures
  • 13 hrs pc labs

Department

Department of Mathematics (UMAT)

Subject specific learning outcomes and competences

The student will obtain the basic knowledge of the methods and usage of linear algebra and geometry and will start learning to use the mathematical software.

Learning objectives

The student will learn to solve the linear systems of equations and get basic knowledge from the matrix theory, the theory of the vector spaces and geometry which is necessary for well-understanding to the related courses. In the laboratory practices the student will learn to use MATLAB software for solving problems of linear algebra.

Prerequisite knowledge and skills

There are no prerequisites

Syllabus of lectures

  • The matrices and the matrix operatins. Linear systems of equations. The reduced echalon form. The Gauss-Jordan elimination.
  • The real n-vectors. The linear independence. The rank of a matrix. The regularity versus singularity of a matrix. The Frobenius Theorem.
  • Homogenious and non-nehomogenious linear systems and their solutions. Systems with the regular matrix. The invertibility of matrices. The calculation of the inverse matrix.
  • Determinants and their properties. The Laplace cofactor expansion, the adjoint matrix and Cramer's rule. The vector product in R^3.
  • The vector spaces and their subspaces. The basis and the dimension. The transition matrix.
  • The linear mapping and its matrix representation. The kernel. Examples.
  • The inner(dot) product. The ortogonalization process. The basis representation of the dot product.
  • The orthogonal projection and the element of the best approximation with examples.
  • The operations with vector spaces. The intersection, the sum and the orthogonal complement.
  • The analytic geometry of the real line the real plane in R^2 a R^3.
  • The eigenvalues and eigenvectors. The similarity of matrices. The diagonal form of the real symmetric matrix. The diagonalization transformation.
  • The quadratic forms and their matrix representation. The definiteness of the quadratic forms.
  • Conic sections and quadratic surfaces. Their matrix representation and its diagonalization transformation of the coordinates. Invariants of the quadratic surfaces.

Syllabus of computer exercises

  • The matrices and the matrix operatins. Linear systems of equations. The reduced echalon form. The Gauss-Jordan elimination. Practising with the software Matlab.
  • The real n-vectors. The linear independence. The rank of a matrix. The regularity versus singularity of a matrix. The Frobenius Theorem. Practising with the software Matlab.
  • Homogenious and non-nehomogenious linear systems and their solutions. Systems with the regular matrix. The invertibility of matrices. The calculation of the inverse matrix. Practising with the software Matlab.
  • Determinants and their properties. The Laplace cofactor expansion, the adjoint matrix and Cramer's rule. The vector product in R^3. Practising with the software Matlab.
  • The vector spaces and their subspaces. The basis and the dimension. The transition matrix. Practising with the software Matlab.
  • The linear mapping and its matrix representation. The kernel. Examples. Practising with the software Matlab.
  • The inner(dot) product. The ortogonalization process. The basis reprezentation of the dot product. Practising with the software Matlab.
  • The orthogonal projection and the element of the best approximation with examples. Practising with the software Matlab.
  • The operations with vector spaces. The intersection, the sum and the orthogonal complement. Practising with the software Matlab.
  • The analytic geometry of the real line the real plane in R^2 a R^3. Practising with the software Matlab.
  • The eigenvalues and eigenvectors. The similarity of matrices. The diagonal form of the real symmetric matrix. The diagonalization transformation. Practising with the software Matlab.
  • The quadratic forms and their matrix reprezentation. The definitness of the quadratic forms. Practising with the software Matlab.
  • Conic sections and quadratic surfaces. Their matrix reprezentation and its diagonalization transformation of the coordinates. Invariants of the quadratic surfaces. Practising with the software Matlab.

Progress assessment

Study evaluation is based on marks obtained for specified items. Minimimum number of marks to pass is 50.

A class accreditation is not defined.

Controlled instruction

Pass the computer practices in the determined range.

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