MATH 461 TOPICS: 

 Systems of Linear Equations
     Gaussian elimination Echelon forms
     Existence and uniqueness of solutions
     Homogeneous systems 
 Matrices 
     Addition, scalar multiplication, multiplication of matrices 
     Elementary matrices and inversion 
     LU decomposition 
     Systems of linear equations as matrix equations 
     Partitioned matrices 
     Determinants and their properties 
     Cramer's rule 
 Vector Spaces 
     Definition and examples 
     Subspaces and spanning sets 
     Linear independence 
     Basis and dimension 
     Row space 
     Column space 
     Rank of a matrix 
     Null space of a matrix 
 Linear Transformations 
     Definition and examples 
     Kernel and range 
     Matrix representation 
     Change of basis and similarity of matrices 
 Scalar Products and Orthogonality 
     Definitions and examples 
     Cauchy-Schwarz inequality 
     Triangle inequality 
     Pythagorean theorem 
     Orthogonal projection 
     Least squares problems 
     Orthonormal sets and orthogonal matrices 
     Gram-Schmidt process and QR factorization 
     Orthogonal polynomials (optional) 
 Eigenvalues 
     Definitions and examples 
     Diagonalization of matrices 
     Spectral theorem for hermitian matrices 
     Quadratic forms 
     Positive definite matrices