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Lecturer(s)
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Pastrňák Robin, Mgr.
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Jukl Marek, doc. RNDr. Ph.D.
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Lachman Dominik, Mgr.
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Vítková Lenka, Mgr. Ph.D.
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Course content
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1. Orthogonal projection, orthogonal homomorphism end isomorphism of euclidean vector space (isometry). 2. Factor vector space. 3. Dual vector space. 4. Endomorphisms, ring and linear algebra of endomorphisms. Similarity of square matrices. 5. Minimal and characteristic polynomials of endomorphism and matrix. 6. Invariant subspaces with respect to endomorphism. Vector subspaces associated with eigenvalues. 7. Root subspaces with respect to endomorphism. The Jordan basis, normal Jordan form of square matrices. 8. Ring of square polynomial matrices. Equivalency of polynomial matrices. 9. System of the greatest common divisors and system of invariant factors of a polynomial matrix, construction of the norma Jordan form of matrix. 10. Bilinear form on a vector space. 11. Quadratic form on a vector space. Polar bilinear form. 12. Conjugated vectors with respect to quadratic forms. Principal directions of quadratic forms in Euclidean vector space. 13. Signature of a quadratic form. Sylvester theorem and Sylvester criterion. 14. Generalised matrix inverse. Moore-Penrose homomorphism.
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Learning activities and teaching methods
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Monologic Lecture(Interpretation, Training), Dialogic Lecture (Discussion, Dialog, Brainstorming)
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Learning outcomes
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1. Orthogonal projection, orthogonal homomorphism end isomorphism of euclidean vector space (isometry). 2. Factor vector space. 3. Dual vector space. 4. Endomorphisms, ring and linear algebra of endomorphisms. Similarity of square matrices. 5. Minimal and characteristic polynomials of endomorphism and matrix. 6. Invariant subspaces with respect to endomorphism. Vector subspaces associated with eigenvalues. 7. Root subspaces with respect to endomorphism. The Jordan basis, normal Jordan form of square matrices. 8. Ring of square polynomial matrices. Equivalency of polynomial matrices. 9. System of the greatest common divisors and system of invariant factors of a polynomial matrix, construction of the norma Jordan form of matrix. 10. Bilinear form on a vector space. 11. Quadratic form on a vector space. Polar bilinear form. 12. Conjugated vectors with respect to quadratic forms. Principal directions of quadratic forms in Euclidean vector space. 13. Signature of a quadratic form. Sylvester theorem and Sylvester criterion. 14. Generalised matrix inverse. Moore-Penrose homomorphism.
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Prerequisites
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unspecified
KAG/LA1A
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KAG/LA1M
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Assessment methods and criteria
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unspecified
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Recommended literature
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Bican, L. (2009). Lineární algebra a geometrie. Praha.
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Hefferon J. (2017). Linear algebra. Colchester.
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Jukl, M. (2000). Bilineární a kvadratické formy. Olomouc.
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Jukl, M. (2006). Lineární algebra. Euklidovské vektorové prostory Homomorfizmy vektorových prostorů. Olomouc.
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Jukl, M. (2001). Lineární operátory. Olomouc.
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Zlatoš, P. (2011). Lineárna algebra a geometria. Bratislava.
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