EXAM 1: Chapters 1-3
Chapter 1: Matrix Arithmetic (matrix addition, scalar multiplication, and matrix multiplication), when is it possible to perform these operations, properties satisfied by these operations (distributive, associative, commutative, ect.), identity (multiplicative, additive), additive inverse, diagonal matrices, square matrices, symmetric and skew-symmetric matrices, transpose.
Chapter 2: Rotation matrices, Orthogonal matrices
Chapter 3: Elementary Row and Column operations, Row-Echelon matrices, Hermite matrices, Normal matrices (N=PAQ), Elementary matrices, linear combination of vectors, linear independence/dependence, row and column rank, row equivalence, equivalence, Coefficient matrices, Augmented Coefficient matrices, Homogeneous and Non-homogeneous systems, consistent and inconsistent systems, the connections between all the concepts listed above.
EXAM 2: Chapters 4-6
Chapter 4: How to tell whether a matrix has an inverse. How to find the inverse of a matrix. Theorem 4.3 is the biggest result from this chapter, know it. How to use inverse matrices to solve a non-homogeneous system of equations.
Chapter 5: Properties of a Vector Space. Whether a subset of a Vector space is a Subspace. Linear Combination of vectors (in the abstract sense). What it means for a subset to span a vector space (or subspace). How to determine whether a set spans a subspace. The definition of Span(S) (what is the span of a subset of vectors). Linear Independence. Basis. How to tell whether a collection of elements is a basis or not. Dimension of a vector space. The Big Theorem we did in class about the size of all bases of a vector space being the same size.
Chapter 6: Linear Transformations. Direct Image. Inverse Image. Properties of direct and inverse images (inclusion preserving, send subspaces to subspaces). The image of a linear map. The kernel of a linear map. Injective. Surjective. Bijective maps. Dimension Theorem. Theorem 6.5,
EXAM 3: Chapters 7-9
Chapter 7: Ordered basis, how to find a transformation matrix (with respect to ordered bases), Change of basis matrix (transition matrices), how to express a transformation matrix with respect to different bases, Similarity.
Chapter 8: How to compute determinants for any general n by n matrices (there are tricks for 2 by 2 and 3 by 3) by cofactor expansion along a row (or column), Compute a determinant by putting in row echelon form via elementary row operations, Minors, Cofactors, Adjoint, How to find an inverse using determinants (and adjoints), How to solve a system of non-homogeneous linear equations using determinants (Cramer's Rule), Properties of determinants (i.e. determinant of a product of matrices = product of determinants, ect.)
Chapter 9: Definition of an eigenvalue of a matrix (which can be zero!), Definition of an eigenvector of a given eigenvalue, Characteristic polynomial, (algebraic) Multiplicity of an eigenvalue, Finding the eigenspace of a given eigenvalue, dimension of the eigenspace (geometric multiplicity), Finding a basis for an eigenspace, Diagonalizable matrices.
FINAL: Everything
The final is Friday May 6th from 5-7 PM in our usual room Psych 309. We will have a review session tomorrow May 5th from 3:30 - 4:30 in Math 102. It is on the first floor of the main Math building (the big one). Basically we will probably discuss some of the practice problems I have on the link below. If you gave me a nicknmame at the last class, check out your final grades going into the final.
Go over everything listed above. BE SURE TO REVIEW NOTES ABOUT MINIMAL POLYNOMIALS. You WILL be asked about them on the final. Below is a link to a bunch of problems to help you prepare for the final.