Overview

The fundamental algebra of vectors and matrices including addition, scaling, and multiplication. Block operations with vectors and matrices. Algorithms for computing the LU (Gauss) factorization of an MxN matrix, with pivoting. Matrix representation of systems of linear equations and their solution via the LU factorization. Basic properties of determinants. Matrix inverses. Linear transformations and change of basis. The four fundamental subspaces and the fundamental theorem of linear algebra. Introduction to eigenvalues and eigenvectors.

Prerequisites

• EC1010 (May be taken concurrently.)

• Students should have mathematical background at the level generally expected of someone with a B.S. in Engineering, i.e., familiarity with calculus and solid algebra skills.

Learning Outcomes

• Determine whether a system of linear equations is consistent and find its general solution.
• Rowreduce  a  matrix to reduced echelon form.
• Carry out matrix operations, including inverses and determinants.
• Demonstrate understanding of linear dependence and independence.
• Demonstrate understanding of subspaces, span, basis and dimension.
• For the four fundamental subspaces associated with a matrix, under- stand their relationships and find bases.
• Find the rank and nullity of a given matrix and understand the rank- nullity theorem.
• Apply Gram-Schmidt process to obtain an orthogonal or orthonormal set.
• Compute QR decomposition.
• Solve least-squares problems.
• Understand determinants and their properties.
• Apply Cramer’s Rule to solve a linear system.
• Determine eigenvalues and eigenvectors of a square matrix.
• Find the diagonalization of a matrix if possible.
• Recognize symmetric matrices, positive definite matrices, orthogonal matrices, and similar matrices and understand their properties.
• Change the coordinates of a vector from one basis to another basis.
• Understand linear transformations and their matrix representations.
• Understand singular value decomposition.