Math Final Review Dec 10, 2010


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1 Math Final Review Dec 10, 2010 General Points: Date and time: Monday, December 13, 10:30pm 12:30pm Exam topics: Chapters 1 4, 5.1, 5.2, 6.1, 6.2, 6.4 There is just one fundamental way to prepare for the final exam: understand the material! To review for the exam, I suggest studying all the midterm exams and the homework problems. Solutions to the midterm exams and selected problems from the homework can be found on the course web page. You will be allowed to prepare and use an 8.5 inch by 11 inch piece of paper with your notes during the exam. You will also be allowed to use a calculator. You will answer questions on the exam itself. There will be a mix of true/false and problems on the exam. To receive full credit on the problems, all work must be shown and correct. When you receive the exam, relax and proceed deliberately. If you don t know how to do a problem, skip it and return to it later. Accuracy is paramount, speed is useless! The exam is not a contest to see who can finish the fastest. If you finish early, be content that you now have time to double and triple check your answers. Chapter 1: 1.1: Vectors and linear combinations 1. Know both the algebraic and geometric view of taking linear combinations of vectors. What do all linear combinations of linearly independent vectors give you? 2. Parallelogram rule 1.2: Lengths and dot products 1.3: Matrices 1. Formula for dot product between two vectors. 2. Geometric meaning of the dot product between two vectors; cosine forumla. 3. How to find the length (norm) of a vector in terms of dot products. 4. Definition of a unit vector and how to compute it. 5. Know how to apply Schwarz inequality: v w v w. 6. Know how to apply Triangle inequality: v + w v + w 1. Two views of matrixvector multiplication Ax: 1) linear combination of columns of A; 2) Dot product of rows of A with x. 2.1: Vectors and linear equations 1. Geometric view of solving linear system of equations Ax = b. (a) Row view: intersecting planes (b) Column view: adding vectors together to produce b. 1
2 2 2.2: Elimination idea 1. Know how to put a linear system Ax = b into upper triangular form so that the system can be easily solved by back substitution. 2. Know the three things that can go wrong in elimination and what they mean in terms of solving the linear system of equations. (a) A pivot=0. (b) Reduced augmented system has a row of all zeros. (c) Reduced augmented system has a row of all zeros except the last column. 2.3: Elimination using matrices 1. Know how to do elimination using matrix products on the augmented system. One can reduce the system to upper triangular form using a sequence of multiplications of the system by lower triangular matrices. 2.4: Rules for matrix operations 1. Know properties for matrix multiplication: associative law, left and right distributive law, identity law. 2. What about a commutative law? 3. Two views of matrix multiplication. 4. Know how many numbers must be multiplied when doing matrixmatrix multiplication and how to intelligently multiply several matrices together. 2.5: Inverse matrices 1. Definition of the inverse of a square matrix. 2. A 1 exists if and only if using elimination to solve Ax = b produces n nonzero pivots. 3. If the inverse of A exists then it is unique. 4. If A is invertible then the solution to Ax = b is x = A 1 b. 5. If A is invertible, then the only solution to Ax = 0 is x = Know how to compute the inverse of a 2by2 matrix. 7. Know how to find the inverse of a diagonal matrix. 8. Rule for splitting up the inverse of a product. 2.6: LU factorization 1. Definitely know how to compute the LU factorization of a matrix. 2. Know how to solve a linear system of equations using LU factorization. 2.7: Transposes and Permuations 1. Properties of the transpose of a matrix. 2. Definition of a symmetric matrix. 3. Know how to compute inner and outer products using the transpose. 3.1: Vector spaces 1. Definition of a vector space and the 8 properties a vector space must satisfy (see page 127). 2. Definition of a subspace of a vector space. 3. Definition of the column space of a matrix and how it relates to Ax = b having a solution. 3.2: Nullspace 1. Definition of the nullspace of a matrix, N(A) 2. Know how to find N(A) using rref.
3 3 3.3: Rank 3. If A is invertible what is N(A). 1. Definition of the rank(a) in terms of pivots, independent rows and columns of A, and the dimension of C(A) and C(A T ). 2. What is the rank of xx T? 3.4: Complete solution to Ax = b 1. Know how to find the complete solution of Ax = b. 2. Definition of matrices with full column or full row rank. 3. Definitely know the relationship between the number of possible solutions to Ax = b and the rank and size of A (4 possibilities). 3.5: Linear independence, basis, and dimension 1. When is a set of vectors linearly independent? 2. What does the span of a set of vectors mean? 3. What do C(A) and C(A T ) mean in terms of the span of some vectors? 4. Know the definition of a basis for a vector space. 5. Know how to find a basis for C(A), C(A T ), and N(A). 6. How is the dimension of a subspace defined in terms of a basis? 3.6: Dimension of the four fundamental subspaces 1. Definition of the four fundamental subspaces of a matrix and the sizes of the vectors in these spaces. 2. Definitely know The Big Picture in Figure 3.5, p Definitely know part I of the fundamental theorem of linear algebra. 4. Know how to find a basis for N(A T ). 4.1: Orthogonality of the four fundamental subspaces 1. What does it mean for two vectors to be orthogonal? 2. What is the geometric picture of two orthogonal vectors? 3. Definition of a orthogonal subspace. 4. Definition of the orthogonal complement of a subspace and how to find it. 5. Part II of the fundamental theorem. 6. If A is mbyn then any vector in R n can be written as x = x r + x n, where x r is in C(A T ) and x n is in N(A). 7. Definitely know the picture displayed in Figure 4.3 showing the true action of a matrix A on a vector x. Where does Ax r go? Where does Ax n go? Can you reproduce this picture if asked? 8. If you have n independent vectors in R n, what is the span of these vectors equal to? 4.2: Projections 1. Geometric picture of the projection of a vector onto a subspace (in two dimensions). 2. How to project onto a line. 3. If V is a subpace and p is the projection of b onto V then b p is?? to any vector from V. 4. Know how to project onto a subspace V. 5. What property guarantees A T A is invertible? 6. You should definitely know how to project a vector onto a subspace. For example, a really good question is the extra credit question from midterm 3. This involves all sorts of things we have learned and the key to solving it is to use projections!
4 4 4.3: Least squares 1. When does Ax = b have a solution? 2. There will definitely be a problem on computing the least squares solution to an overdetermined system. 4.4: Gramschmidt and orthogonal bases 1. Definition of a set of orthogonal and orthonormal vectors. 2. Nice property of matrices with orthonormal columns (in the case of square matrices we call these orthogonal matrices). 3. Know how to do projections with orthogonal bases. 4. Given a set of vectors, know how to use GramSchmidt to convert the set to an orthonormal set that spans the same space. 5. Know how to compute the QR decomposition of a matrix. 6. Know the technique for solving least squares problems with QR decomposition. 5.1: Determinants 1. Know formula for calculating the determinant of a 2by2 matrix. 2. Know the 10 properties regarding the determinant given on pp and how to apply them. 5.2: Permutations and cofactors 1. Know how to find the determinant of A from the P T LU decomposition of A. 2. Know how to compute the determinant of a matrix using cofactors. 6.1: Eigenvalues and eigenvectors 1. Know how to find the eigenvalues and eigenvectors of a matrix using the characteristic equation and rref. 2. Know the geometric meaning of an eigenvalue, eigenvector of a matrix. 3. Know the relationship between the determinant of a matrix and its eigenvalues. 4. Know the relationship between the trace of a matrix and its eigenvalues. 5. Relationship between the eigenvalues and eigenvectors of A and A 2, A 3,..., A k (k > 1). 6. Relationship between inverse of A and its eigenvalues and eigenvectors. 6.2: Diagonalizing a matrix 1. Know the theorem regarding the diagonalization of a matrix. What are the assumptions? 2. Know how to diagonalize a matrix. 3. How can the powers of A be computed easily from the eigenvector diagonalization? 4. Properties of diagonalization (a) If A has n independent eigenvalues then the eigenvectors are independent. So A can be diagonalized. (b) The eigenvectors of S come in the same order as the eigenvalues of Λ. (c) If x is an eigenvector then so is cx where c is any nonzero constant. (d) If A does not have n linearly independent eigenvectors then it cannot be diagonalized. (e) If A has a zero eigenvalue then A is not invertible. It may still be diagonalizable (if its n eigenvectors are independent). 5. If you know the eigenvalues and eigenvectors of A then what are the eigenvalues and eigenvectors of A + ci, where c is a constant and I is the identity matrix. 6. Pitfalls:
5 5 (a) If λ is an eigenvalue of A and β is an eigenvalue of B then λβ is not generally an eigenvalue of AB. (b) Also λ + β is not generally an eigenvalue of A + B. 6.4: Symmetric matrices 1. A symmetric matrix has only real eigenvalues. 2. A symmetric matrix has orthonormal eigenvectors. 3. If A is symmetric then it can be diagonalized as A = QΛQ T, where Q is orthogonal and contains the eigenvectors of A in its columns. 4. Every symmetric matrix can be diagonalized. See the conceptual question review starting on page 552 of the book.
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