Lecture Notes
Lecture 0 (01/10/2022)
Introduction - linear equations and their solutions.
Lecture 1 (01/12/2022)
Matrices. Coefficient and augmented matrices of a system of linear equations, Echelon form.
Lecture 2 (01/14/2022)
Reduced echelon form, Gauss-Jordan algorithm, consistent vs inconsistent systems, row equivalent matrices.
Lecture 3 (01/19/2022)
Summary. Number of solutions. Homogeneous systems.
Lecture 4 (01/21/2022)
Matrices: algebraic operations.
Lecture 5 (01/24/2022)
More on matrix multiplication. Solutions of a linear system in vector form.
Lecture 6 (01/26/2022)
Formal properties of matrix operations. Transpose of a matrix.
Lecture 7 (01/28/2022)
Invertible matrices. Computing inverse via Gauss-Jordan.
Lecture 8 (01/31/2022)
Linear independence. Non-singular=Invertible matrices. Rank of a matrix.
Lecture 9 (part 1) (02/02/2022)
Summary.
Lecture 9 (02/02/2022)
Vectors in 2D and 3D. Addition, scalar multiplication, norm.
Lecture 10 (02/04/2022)
Dot product. Cosine formula. Projection formula.
Lecture 11 (02/09/2022)
Cross product. Sine formula. Applications.
Lecture 12 (02/11/2022)
Equations of lines and planes.
Lecture 13 (02/14/2022)
Vector space R^n and its subspaces.
Lecture 14 (02/16/2022)
Null space and range of a matrix.
Lecture 15 (02/18/2022)
Basis and dimension of a subspace.
Lecture 16 (02/21/2022)
Methods of finding a basis.
Lecture 17 (02/23/2022)
Rank-nullity theorem. Orthogonal and orthonormal basis.
Lecture 18 (02/25/2022)
Gram-Schmidt algorithm. Linear transformations.
Lecture 19 (02/28/2022)
Linear transformation continued. Matrices as linear transformations.
Lecture 20 (03/02/2022)
Linear transformations: dependence on the choice of bases.
Lecture 21 (03/02,09/2022)
Abstract vector spaces: introduction.
Lecture 22 (03/09,11/2022)
Subspaces. Span of a set of vectors. Linear independence. Basis.
Lecture 23 (03/11/2022)
Properties of basis. Coordinates relative to a basis.
Lecture 24 (03/21/2022)
Linear transformations. One-one, onto, isomorphisms. Null space and range.
Lecture 25 (03/23/2022)
Invertible linear transformations. Matrix representation of a linear transformation.
Lecture 26 (03/25/2022)
Matrix representation continued.
Lecture 27 (03/28/2022)
Definition of determinant. Basic properties.
Lecture 28 (03/30/2022)
Change in determinant under row operations.
Lecture 29 (04/01/2022)
Cofactor matrix. Inverse of a matrix. Cramer's rule.