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 :

1. Why Linear Algebra?

2. Linear Systems of Equations, Gaussian Elimination

3. Matrices

4. Rank, Trace and the Determinant of a matrix. These are important invariants in Linear Algebra

5. Vector spaces and sub-vector spaces

6. Basis, dimension, linear dependence/independence, spanning sets and span

7. Important 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 transformation

8. Linear transformations. Conditions of a linear transformation to be injective, surjective, bijective

9. Relation between matrices and linear transformations. Coordinates, Matrices representing a linear transformation

10. Dimension theorems - This is a very important and powerful topic

11. Eigenvalues, Eigenvectors and Diagonalization

12. Inner 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 context

Happy Linear Algebra !