## Spring 2018

## MATH410

## Linear Algebra and Financial Applications

### Syllabus:

This course emphasizes techniques of linear algebra and applications to actuarial science, finance, economics, and statistics. Topics include vector and matrix theory, systems of linear equations, inverses and determinants, vector spaces and subspaces, orthogonality and projections, eigenvalues and eigenvectors, modern portfolio theory, and linear regression models. R programming is integrated to each topic.

### Learning Objectives:

**Chapter 1: Introduction to Vectors and Matrices** - Vectors and Vector Operations; Matrices and Matrix Operations; Mathematical Model in Modern Portfolio Theory.

**Chapter 2: Systems of Linear Equations** - System of Linear Equations; Gaussian Elimination; Solution of Mean-Variance Portfolio Analysis.

**Chapter 3: Inverse Matrices and Determinants** - Elementary Matrices; Matrix Inversion; Gauss-Jordan Elimination; Determinants; Further Analysis for Solution of Mean-Variance Portfolio Analysis.

**Chapter 4: Vector Spaces and Subspaces** - Vector Spaces; Subspaces and Spans; First Important Subspace: Column Space; Second Important Subspace: Nullspace; Rank and Complete Solution of System of Linear Equations; Independence and Basis; Two More Subspaces and Dimension.

**Chapter 5: Orthogonality and Projection** - Inner Product, Norm, Metric, and Orthogonality; Orthogonality of Four Important Subspaces; Least Squares Approximations and Orthogonal Projections; Linear Regression Models; Orthogonal Basis and Gram-Schmidt.

**Chapter 6: Eigenvalues and Eigenvectors** - Eigenvalues and Eigenvectors; Diagonalization; Homogeneous Markov Chain.

### References:

1. Strang, G. (2016). Introduction to Linear Algebra, 5th edition, Wellesley-Cambridge Press.

2. Lay, D. C., Lay, S. R., and McDonald, J. J. (2015). Linear Algebra and Its Applications, 5th edition, Pearson.

3. Leon, S. (2015). Linear Algebra with Applications, 9th edition, Pearson.

4. Luenberger, D. G. (2013). Investment Science, 2nd edition, Oxford University Press.

5. Cochrane, J. H. (2005). Asset Pricing, revised edition, Princeton University Press.

6. James, G., Witten, D., Hastie, T., and Tibshirani, R. (2013). An Introduction to Statistical Learning with Applications in R, Springer.