MATH 54 FINAL EXAM STUDY GUIDE
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1 MATH 54 FINAL EXAM STUDY GUIDE PEYAM RYAN TABRIZIAN This is the study guide for the final exam! It says what it does: to guide you with your studying for the exam! The terms in boldface are more important than others, so make sure to study them in detail! Note: Remember that the final exam is cumulative. Suggestion: Start with the differential equations-part, since it s fresh in your mind. Once you re done with that, turn your attention to the linear algebra-part. DIFFERENTIAL EQUATIONS Chapter 4: Linear Second-order equations Find the general solution to a second-order differential equation, possibly including complex roots, repeated roots, or initial conditions (4.2.1, 4.2.5, , 4.3.5, , ) Determine if two functions are linearly independent or linearly dependent (4.2.27, ) Solve equations using undetermined coefficients (4.4.14, , , , ) Solve equations using variation of parameters (4.6.3, ) Chapter 6: Theory of higher-order linear differential equations Find the largest interval on which a differential equation has a unique solution (6.1.3, 6.1.5) Determine if a set of functions is linearly independent or linearly dependent (6.1.9, , , ) Show that given functions form a fundamental solution set for a differential equation (6.1.15, , ) Date: Tuesday, May 12,
2 2 PEYAM RYAN TABRIZIAN Find the general solution of a higher-order differential equation, possibly including initial conditions (6.2.7, , , , , ); know the technique using the rational roots theorem and long-division. Chapter 9: Matrix methods for linear systems Convert a higher-order equation into a system of differential equations (9.1.13) Determine if a set of vector functions is linearly independent or linearly dependent, using the Wronskian (9.4.16, ) Determine if vector functions form a fundamental solution set for x = Ax Find the general solution to x = Ax, possibly including complex eigenvalues (9.5.13, , , , , 9.6.1, 9.6.9, ) Solve x = Ax, where A is not diagonalizable (9.5.35, ) Solve x = Ax + f using undetermined coefficients. (9.7.3, 9.7.9, ) Note: Remember that variation of parameters for systems will not be on the exam Find e At : - By diagonalizing A (9.8.9) - By using the definition and the fact that (A λi) k = 0 for some k (this only works when A has only one eigenvalue and one eigenvector) (9.8.5) - By using generalized eigenvectors (9.8.11) Use e At to solve x = Ax (9.8.21) Chapter 10: Partial differential equations Calculate the Fourier series of a function f on a given interval, and determine to which function that Fourier series converges (10.3.9, , , , , ) Calculate the Fourier cosine/sine series for a function f, and determine to which function that Fourier series converges to (10.4.5, , ). Using separation of variables, solve the heat/wave/laplace s equation, subject to various boundary/initial conditions ( , ,
3 MATH 54 FINAL EXAM STUDY GUIDE , , , , , , , ). You ARE responsible for heat/wave/laplace s equations with inhomogeneous boundary conditions (see ) and inhomogeneous heat/wave equations (see and the example Prof. Vojta did on page 4 of his May 1-notes), but first focus on solving the problems above, as they are more important. Note: Remember that the following Chapter 10 topics will NOT be on the final exam: - Heat equation in higher dimensions (Example 4 in 10.5) - Traveling waves/d Alembert s formula (pages in 10.6) - Laplace s equation on a disk (for section 10.7, you re only responsible for Example 1) LINEAR ALGEBRA Suggestion: Ignore chapters 1, 2, 3, because you basically already know how to do them! Instead, focus on chapters 4, 5, 6, 7 Chapter 4: Vector Spaces Determine if a set is a vector space or not (4.1.3, 4.1.9, 4.2.7, 4.2.9) Determine whether a set is linear independent or dependent (4.3.3, ) Find a basis and state the dimension of a vector space (4.5.7, ) Given a matrix A, find a basis for Nul(A), Col(A), Row(A), and also find Rank(A) (4.5.15, 4.6.1, 4.6.3) Use the rank-nullity theorem to find Rank(A) etc. (4.6.5, 4.6.9, , , ) Find the change-of-coordinates matrix from B to C (4.4.1, 4.4.3, 4.4.5, 4.4.7, , 4.7.5, 4.7.7, 4.7.9) Use the change-of-coordinates matrix to find [x] C [x] B (4.7.1, 4.7.3); also know how to do this for polynomials (4.7.13)
4 4 PEYAM RYAN TABRIZIAN Chapter 5: Diagonalization Don t focus too much on this chapter, because it s almost the same as Chapter 9. Find a diagonal matrix D and a matrix P such that A = P DP 1, or say A is not diagonalizable (5.2.9, , 5.3.9, , ) Show that a given matrix is not diagonalizable (5.3.8, ) Find the matrix of a linear transformation (5.4.1, 5.4.3, 5.4.9, , (b)); also know how to do this for more abstract examples, like matrices or polynomials (see for example Peyam s midterm 2) Note: We will not ask about complex eigenvalues, except maybe in the form of a Chapter 9 - question! Chapter 6: Inner Products and Norms Find a basis for W (all you do is find a basis for W, and find all vectors which are perpendicular to the vectors in the basis, just like the question above) Find the orthogonal projection of x on a subspace W. Use this to write x as a sum of two orthogonal vectors, and to find the smallest distance between x and W (6.2.11, 6.3.3, 6.3.5, 6.3.1, 6.3.7, , ) Use the Gram-Schmidt process to produce an orthonormal basis of a subspace W spanned by some vectors (6.4.9, ) Find the QR-factorization of a matrix (6.5.13, ). Find the least-squares solution (and least-squares error) of an inconsistent system of equations (6.5.1, 6.5.3, 6.5.7, 6.5.9, ) Find inner products, lengths, and orthogonal projections of functions f and g using fancier inner products f, g (6.7.3, 6.7.5, 6.7.7, 6.7.9, , , ) Use the Gram-Schmidt process to find an orthonormal basis of functions (6.7.25, ) Remember the Cauchy-Schwarz inequality (6.7.19, ) Chapter 7: Symmetric matrices and quadratic forms Given a symmetric matrix A, find an diagonal matrix D and an orthogonal matrix P such that A = P DP T (7.1.13, , )
5 MATH 54 FINAL EXAM STUDY GUIDE 5 CONCEPTS Understand the following concepts: - Pivots (1.2, 1.5) - Span (1.3, 1.4) - Linear independence (1.7) - One-to-one and onto (1.9) - Invertible matrices (2.2) - Know all the implications of the Invertible Matrix Theorem (2.3, including page 144) - Subspace, Basis (2.6) - Nul(A), Col(A), The Rank Theorem (2.6, 4.6) - Vector space, Subspace (4.1) - Basis, Dimension (4.3, 4.5) - Coordinates of x with respect to B (4.4) - Basis, Dimension (4.5) - Change of coordinates matrix (4.7) - Eigenvalues, Eigenvectors, Characteristic polynomial ( ) - A is similar to B (5.2) - Diagonalizable, Diagonalization Theorem (Theorem 5 in section 5.3) - Matrix of a Linear transformation (5.4) - Inner products, Norms, Orthogonal vectors, Orthogonal Matrix (6.1) - W (6.1) - (Row(A)) = Nul(A), (Col(A)) = Nul(A T ) (Theorem 3 in 6.1) - Orthogonal projection (6.2, 6.3) - Least-squares solution (6.4) - Inner product, inner product space (6.7) - Cauchy-Schwarz and Triangle Inequality (6.7) - Orthogonally diagonalizable (7.1) - Existence and Uniqueness Theorem for differential equations (6.1) - Linear independence of functions/vector functions and Wronskian/fundamental matrix (4.2, 6.1, 9.4) - Fundamental solution set (6.1, 9.4) - Exponential of a matrix (9.8) - Separation of variables (10.2) - Fourier series, Fourier cosine/sine series (10.3, 10.4) - Pointwise convergence of Fourier series (10.3) - Orthogonal expansions (10.3) - Maximum principle (10.5), Conservation of Energy (10.6)
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