Exam 1 study guide, Winter, 2016
Given two vectors u and v, calculate their dot product.
Find the angle between two given vectors.
Find a vector or line perpendicular to a given vector.
Given two matrices, find their product.
Understand matrix multiplication either as linear combination of columns or of rows.
Find the inverse of a given 2 by 2 or 3 by 3 matrix using Gauss-Jordan elimination, and show work.
Understand and use linear combinations.
Identify the elementary matrix which accomplishes a given row operation.
Given a directed graph, find its adjacency matrix, and vice versa, and use the matrix to find the number of paths of a given length.
Know how to row reduce a matrix to U form, or to reduced echelon form.
Find the LU or LDU decomposition of a matrix.
Understand what a symmetric matrix is.
Know how to find the transpose or inverse of a product of two matrices in terms of the transposes and inverses of the two matrices, that is (AB)^(-1) = B^(–1)A^(–1) and (AB)^(t) = B^(t)A^(t).
Determine whether a set of vectors or matrices is a subspace of a larger vector space.
Determine whether a given set of linear equations has a solution.
Identify and express the column, row, and null space of a matrix.
Find and express all solutions to Ax=b in parameter form.
What are the effects of matrix multiplication by a 2 by 2 matrix on the unit square in the first quadrant?

Vocabulary for exam 1
1. Non- Singular or invertible
2. Singular or non-invertible
3. Pivot
4. Row-echleon form
5. Reduced row-echleon form
6. Inconsistent system
7. Vector Space
8. Vector Subspace
9. Linear Combination
10. Augmented Matrix
11. Leading Variable
12. Null Space or Kernel
13. Parameter
14. Free Variable
15. Row space
16. Column Space
17. Inverse
18. Transpose
19. Symmetric matrix
20. Diagonal Matrix
21. LU and LDU form
22. Upper or lower triangular matrix

Study guide exam 3, sections 4.3 to 6.6, Fall 2015
How to do a least squares approximation
How to use an orthogonal basis
Gram-Schmidt process
Use the determinant properties
Determinant calculation methods: permutations, co-factors, row-reduction
Cramer’s rule for solving Ax=b
Finding the inverse using determinant and cofactors
Using determinant to find volume
How to calculate eigenvalues
Use eigenvalues to diagonalize a matrix
Use eigenvalues and diagonalized form to find powers of a matrix (as with the Fibonacci numbers)
Markov Matrices
Symmetric matrices (eigenvalues are real, symmetric matrices are diagonalizable, eigenvectors of distinct eigenvalues are orthogonal, etc.)
Similar matrices (have same rank, same eigenvectors, same trace and determinant, etc.)
(We have not covered section 6.5 or Jordan form in section 6.6; we have not spent much time on differential equations.)

Study Guide for Math 2B final exam, Fall 2015
(1) Given an m by n matrix A find its
Null space and nullity
Column space, row space, and rank
Inverse, if one exists
Determinant, if it is square.
(2) The four important spaces: column and row spaces of A and At and their null spaces, and which are orthogonal to which.
(3) Find the eigenvalues and eigenvectors of matrix A
(4) Diagonalize matrix A, if possible
(5) Given an eigenvalue of a matrix, describe its eigenspace
(6) Given an orthonormal basis, express a vector in that basis.
(7) Find the dot product or angle between two vectors, and the length of a vector.
(8) Rotate a vector in R2, or find its reflection in one of the axes.
(9) How do you know if a set of vectors is a subspace?
(10) When are unions or intersections of subspaces also subspaces?
(11) Given a set of vectors, determine if they are linearly independent.
(12) Given a set of vectors, determine if they span a space.
(13) Express a vector in one basis in terms of any other basis.
(14) Find transition matrices from one basis to another
(15) Understand geometric properties of the dot product.
(16) Give a matrix for a linear transformation
(17) Reflection-, rotation-, shear-, dilation-matrices in R2.
(18) Various examples of vector spaces, such as that of matrices, that of polynomials, etc
(19) Use graphs expressed via adjacency matrices to find number of paths, etc.
(20) Solve a matrix equation.
(21) Find a matrix for a projection, find the projection of one vector on another.
(22) Basic ideas about Markov matrices and Google page rank.
(23) Use Gram-Schmidt to find an orthogonal basis.

Some theorems:
(23) Isometries have +1 or –1 as only eigenvalue(s).
(24) Determinant of isometry matrices have value 1 or –1.
(25) Isometries in the plane are reflections or rotations.
(26) Every square matrix may be expressed in LDU form uniquely.
(27) (AB)-1 = B-1A-1, (AB) )t = BtAt
(28) If a square matrix is singular one of its eigenvalues must be zero
(29) If the eigenvalues are distinct, then the eigenvectors are independent.
(30) The eigenvectors for distinct eigenvalues for a symmetric matrix are orthogonal.
(31) Similar matrices have the same eigenvalues, but not the same eigenvectors.
(32) The dimension of the set of all m by n matrices, as a vector space with respect to matrix addition and scalar muiltiplication, is mn.
(33) The sum of the eigenvalues = trace, product = determinant.
(34) Projection matrices have eigenvalues 1 or 0.
(35) A and A-1 have same eigenvectors but reciprocal eigenvalues.
(36) A symmetric matrix has real eigenvalues.