Study

# Computing

## Linear Algebra

• Class 60
• Practice 15
• Independent work 75
Total 150

Linear Algebra

Obligatory

183355

1

5

### Course objectives

Matrices. Basic types. Operations with matrices. Inverse matrix. Regular and singular matrices.
Determinants. Basic properties, computation of determinants. Laplace's rule; Elementary transformations, rank of a matrix, linear independence and rank. Characterization of regular matrices.
Elementary transformations, rank of a matrix, linear independence and rank. Characterization of regular matrices; Characterization of regular matrices using determinant. Computation of the inverse matrix.
Gaussian elimination method. Homogeneous and nonhomogeneous systems; Rank of a system and rank of extended matrix.
Rank of a system and rank of extended matrix; Cramer's rule. Comparison of computational complexity of the Gauss and Cramer algorithms.
Operations with vectors. Linear dependence and independence in V^2 and V^3. Bases in V^2 and V^3. Canonical base. Vectors in coordinate systems; Dot product. Norm, ortogonality. Vector and scalar projections of one vector on another.
Cross product of two vectors and mixed product of three vectors; Radius-vector. Coordinates of the midpoint of a segment and of the barycenter of a triangle. Convex combinations of vectors. Convex hull. Convex set.
Midterm exam.
Equations of plane. Mutual position of two planes. Distance between a point and a plane.
Line, parametric and canonical equations. Mutual position of a line and a plane.
Vector spaces and their subspaces. Linear hull (span). The space R^n; Linear independence and dependence. Basis and dimension.
Coordinate system. Basis change. Transition matrix; Inner product. Inner product space. Orthogonal basis. Fourier coefficient. Orthogonal projection. Normed vector space.
Linear operators and their matrix representation. Basis change. Similar matrices; Examples of linear operators in V^2 and V^3. Linear functional. Hyperspace, half-space.
Eigenvalues and eigenvectors. Eigenspaces. Characteristic polynomial. Hamilton-Cayley's theorem; Schur's theorem. Matrix functions. Spectral mapping theorem.
Final exam.

Neven Elezović (2016.), Linearna algebra, Element, Zagreb
N. Elezović, A. Aglić Aljinović (2006.), Linearna algebra: zbirka zadataka, Element, Zagreb
Gilbert Strang (2006.), Linear Algebra and its Applications, Brooks Cole
Damir Bakić (2008.), Linearna algebra, Školska knjiga, Zagreb
David S. Watkins (2002.), Fundamentals of Matrix Computations, Wiley

#### Minimal learning outcomes

• Describe and apply linear algebra basic concepts and methods
• Demonstrate fundamental skills of matrix calculus and solving linear systems of equations
• Apply fundamental knowledge of vector analysis and space analytic geometry
• Demonstrate basic knowledge of vector spaces and linear operators
• Demonstrate an ability to express mathematical ideas and abstract thinking in linear algebra
• Demonstrate an ability to basic problem solving and reaching conclusions in linear algebra
• Use methods of linear algebra in engineering