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Systems of linear equations, Matrices, Elementary row operations, Row-reduced echelon matrices. Vector spaces, Subspaces, Bases and dimension, Ordered bases and coordinates.

Linear transformations, Rank-nullity theorem, Algebra of linear transformations, Isomorphism, Matrix representation, Linear functionals, Annihilator, Double dual, Transpose of a linear transformation.

Characteristic values and characteristic vectors of linear transformations, Diagonalizability, Minimal polynomial of a linear transformation, Cayley-Hamilton theorem, Invariant subspaces, Direct-sum decompositions, Invariant direct sums, The primary decomposition theorem, Cyclic subspaces and annihilators, Cyclic decomposition, Rational, Jordan forms.

Inner product spaces, Orthonormal bases, Gram-Schmidt process.

 Lectures Topic 1 Systems of linear equations, equivalent systems, matrices 2 Elementary row operations, properties, examples 3 Row-reduced echelon matrices, elementary matrices, 4 Invertible matrices, elementary matrices, equivalence 5 Other equivalent conditions in terms of systems of linear equations, homogeneous, non-homogeneous, examples 6 Vector spaces, definition and examples 7 Subspaces, examples 8 Linear independence and dependence, examples 9 Basis, dimension, examples 10 Ordered basis, examples, coordinates 11 Linear transformations, examples 12 Range space, null space, examples 13 Injective, Surjective linear transformations, examples 14 Rank-nullity theorem, consequences, examples 15 Algebra of linear transformations 16 Isomorphism, examples 17 Matrix representation of linear transformations, examples 18 Change of bases, matrix representation, examples 19 Linear functionals, annihilator 20 Double dual, examples 21 Dual basis, properties, examples 22 Transpose of a linear transformation, properties 23 Characteristic Values and characteristic vectors, examples 24 Properties of characteristic values, vectors, examples 25 Similarity transformation, Diagonalizability, characterization 26 Minimal polynomial, properties and relationship with the characteristic polynomial 27 Cayley-Hamilton theorem, applications 28 Invariant subspaces, characteristic values, examples 29 Direct-sum decomposition, examples, properties 30 Invariant direct-sum decomposition, examples 31 The Primary decomposition Theorem I 32 The Primary decomposition Theorem II 33 Cyclic subspaces, annihilators, examples 34 Cyclic decomposition Theorem I 35 Cyclic decomposition Theorem II 36 Rational form, examples 37 Jordan form, examples 38 Inner product spaces, examples 39 Cauchy-Schwarz inequality, examples 40 Orthonormal bases, Gram-Schmidt procedure.
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