Study Guide for the Exam

• do matrix algebra
• understand the technique of row reduction and it's relationship to multiplication by elementary matrices
• understand the properties of the dot and cross product
• use orthogonality and the parametric and normal equations of a plane in 3-space
• prove (or disprove) that a given subset of a vector space is a vector subspace
• understand linear independence, spanning and dimension
• work with vector spaces other than Rⁿ such as spaces of functions
• know how to find the basis for the row space, null space, column space and transpose null space of a matrix A
• solve matrix equations, especially Ax = b and how this relates to the four fundamental spaces of a matrix
• compute the determinant of a 4x4 or larger matrix by using the properties of determinants
• find the eigenvalues and eigenvectors of a matrix A and how this relates to the properties of A
• determine whether a matrix is similar to a diagonal matrix
• determine whether 2 matrices are similar
• find the least squares solution to a curve fitting problem
• understand projections and why least squares solutions are 'best fit' solutions