Linear Algebra: A Problem Based Approach

The focus of this course is on solving problems. Where the best way to benefit from the course is to ask questions and in hand I will respond with answers involving exercises that expand upon the questions. The topics covered are:Why Linear Algebra.

Linear Systems of Equations, Gaussian EliminationMatricesRank, Trace and the Determinant of a matrix. These are important invariants in Linear AlgebraVector spaces and sub-vector spacesBasis, dimension, linear dependence/independence, spanning sets and spanImportant vector spaces: Null space of a matrix, row and column spaces of a matrix, Span of a set, intersection, sum and direct sum of vector spaces, eigenspace, orthogonal complement, Kernel and Image of a linear transformationLinear transformations. Conditions of a linear transformation to be injective, surjective, bijectiveRelation between matrices and linear transformations.

Coordinates, Matrices representing a linear transformationDimension theorems - This is a very important and powerful topicEigenvalues, Eigenvectors and DiagonalizationInner product spaces, norms, Cauchy-Schwartz, general law of cosines - An inner product space is a vector space along with an inner product on that vector space. When we say that a vector space V is an inner product space, we are also thinking that an inner product on V is lurking nearby or is obvious from the contextThe course is highly dynamic and content is uploaded regularly. Happy Linear Algebra.