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Math 54, Exams of Algebra

Rice UniversityAlgebra

Note: Remember that the final exam is cumulative. It contains 3 linear algebra ques- tions and 4 differential equations questions. Also, ...

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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. It contains 3linear algebra ques-
tions and 4differential equations questions. Also, remember that GrĀØ
unbaumā€™s coupled
harmonic oscillators-question will be on the final for sure!
Suggestion: Start with the differential equations-part, since itā€™s fresh in your mind.
Once youā€™re done with that, turn your attention to the linear algebra-part.
DIFFE RE NT IA L EQUAT ION S
Suggestion: Even though the book spends a LOT of time with chapters 4and 6, they
are actually not important! When I took Math 54, I didnā€™t even have questions from those
two chapters. Thatā€™s why, if I were you, I would just briefly review them, and focus my
attention to chapters 9and 10, which are WAY more important!
Chapter 4: Linear Second-order equations
ā€¢Find the general solution to a second-order differential equation, possibly
including complex roots, repeated roots, or initial conditions (4.2.1, 4.2.11,
4.2.15, 4.3.1, 4.3.3)
ā€¢Determine if two functions are linearly independent or linearly dependent (4.2.27,
4.2.29)
ā€¢Solve equations using undetermined coefficients, but do not spend too much time
on this (4.4.9, 4.4.11, 4.4.15, 4.4.17, 4.5.29, 4.9.3)
ā€¢Solve equations using variation of parameters (4.6.1, 4.6.3, 4.6.15)
ā€¢Find the equation motion of a simple harmonic oscillator (4.8.1, 4.8.11)
Chapter 6: Theory of higher-order linear differential equations
ā€¢Find the largest interval on which a differential equation has a unique solu-
tion (6.1.1, 6.1.3, 6.1.5)
ā€¢Determine if a set of functions is linearly independent or linearly dependent
(6.1.9, 6.1.11, 6.1.13, 6.2.25)
Date: Tuesday, December 6th, 2011.
1
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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. It contains 3 linear algebra ques- tions and 4 differential equations questions. Also, remember that GrĀØunbaumā€™s coupled harmonic oscillators-question will be on the final for sure!

Suggestion: Start with the differential equations-part, since itā€™s fresh in your mind. Once youā€™re done with that, turn your attention to the linear algebra-part.

DIFFERENTIAL EQUATIONS

Suggestion: Even though the book spends a LOT of time with chapters 4 and 6 , they are actually not important! When I took Math 54 , I didnā€™t even have questions from those two chapters. Thatā€™s why, if I were you, I would just briefly review them, and focus my attention to chapters 9 and 10 , which are WAY more important!

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, 4.2.11, 4.2.15, 4.3.1, 4.3.3)
  • Determine if two functions are linearly independent or linearly dependent (4.2.27, 4.2.29)
  • Solve equations using undetermined coefficients, but do not spend too much time on this (4.4.9, 4.4.11, 4.4.15, 4.4.17, 4.5.29, 4.9.3)
  • Solve equations using variation of parameters (4.6.1, 4.6.3, 4.6.15)
  • Find the equation motion of a simple harmonic oscillator (4.8.1, 4.8.11)

Chapter 6: Theory of higher-order linear differential equations

  • Find the largest interval on which a differential equation has a unique solu- tion (6.1.1, 6.1.3, 6.1.5)
  • Determine if a set of functions is linearly independent or linearly dependent (6.1.9, 6.1.11, 6.1.13, 6.2.25)

Date: Tuesday, December 6th, 2011. 1

2 PEYAM RYAN TABRIZIAN

  • Find the general solution of a higher-order differential equation, possibly in- cluding initial conditions (6.2.1, 6.2.5, 6.2.9, 6.2.15, 6.2.17, 6.2.19)

Chapter 9: Matrix methods for linear systems

  • Determine if a set of vector functions is linearly independent or linearly de- pendent (9.4.15, 9.4.17)
  • Determine if vector functions form a fundamental solution set for xā€²^ = Ax (9.4.19, 9.4.21)
  • Use the fundamental matrix to find a solution to a given initial-value problem (9.4.27)
  • Find the general solution to xā€²^ = Ax, possibly including complex eigenvalues (9.5.13, 9.5.15, 9.5.19, 9.5.21, 9.5.31, 9.5.33, 9.6.1, 9.6.3)
  • Find the proper frequencies and proper modes of a coupled harmonic oscil- lator (see ā€˜Harmonic Oscillatorā€™-Handout on my website)
  • Solve inhomogeneous systems using undetermined coefficients (9.7.3, 9.7.5)
  • Solve inhomogeneous systems using variation of parameters (9.7.11, 9.7.13, 9.7.15)

Chapter 10: Partial differential equations

  • Calculate the Fourier series of a function f on a given interval, and deter- mine to which function that Fourier series converges (10.3.9, 10.3.17, 10.3.11, 10.3.19, 10.3.13, 10.3.21)
  • Calculate the Fourier cosine/sine series for a function f , and determine to which function that Fourier series converges to (10.4.5, 10.4.7, 10.4.9, 10.4.13, for the second part, you need to understand oddification and evenification, so see 10.4.1, 10.4.3)
  • Using separation of variables, solve the heat, wave, and Laplace equation, subject to various boundary/initial conditions (10.5.1, 10.5.3, 10.5.5, 10.5.7, 10.5.11, 10.6.1, 10.6.3, 10.6.5, 10.6.13, 10.6.15, 10.7.1, 10.7.3, 10.7.5) Note: Do not waste your time reading sections 10.5, 10.6, 10.7, they con- tain tons of useless material! Just know how to solve the problems! Also, donā€™t worry about inhomogeneous heat equations (10.5.9, 10.5.13), or about the Dirich- let problem on a disk (10.7.7)

Less important stuff

  • Sketch trajectories of solutions to xā€²^ = Ax (9.6.17)
  • Calculate eAt, by either diagonalizing A or using the Cayley-Hamilton theorem (9.8.1, 9.8.5, 9.8.7)
  • Show that a given set is orthonormal, and calculate the generalized Fourier series for a given function f in terms of the orthonormal set (10.3.26, 10.3.27)

4 PEYAM RYAN TABRIZIAN

TRUE/FALSE EXTRAVAGANZA Do the following set of T/F questions: 1.5.24, 1.7.21, 1.7.22, 1.9.23, 1.9.24, 2.1.15, 2.8.21, 3.2.27, 4.1.24, 4.2.25, 4.3.21, 4.6.17, 4.7.11, 5.3.21, 6.3.21, 6.5.17 (check out the hints to HW 1-8 for answers)

CONCEPTS

Understand the following concepts:

  • Pivots (1.2.23, 1.2.24, 1.2.25, 1.2.26, 1.5.29, 1.5.31)
  • Span (1.3.22, 1.3.25, 1.4.17, 1.4.29, 1.4.34)
  • Linear independence (1.7.33, 1.7.34, 1.7.35, 1.7.36)
  • Invertible matrices (2.1.23, 2.1.24, 2.2.11, 2.2.19, 2.2.21)
  • Implications of invertibility (2.3.11, 2.3.15, 2.3.17, 2.3.21, 2.3.24, 2.3.30)
  • Subspace, Basis (2.8.1, 2.8.3, 2.8.5, 2.8.7, 2.8.17)
  • Vector space, Subspace (4.1, 4.2)
  • Basis, Dimension (4.3, 4.5)
  • Coordinates of x with respect to B (4.4)
  • N ul(A), Col(A), Row(A), Rank (4.6)
  • Rank-Nullity Theorem (4.6)
  • Change of coordinates matrix (4.7)
  • Eigenvalues, Eigenvectors, Characteristic polynomial (5.1 - 5.3)
  • 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)
  • Orthogonal projection (6.2, 6.3)
  • Gram-Schmidt process (6.4)
  • Least-squares (6.5)
  • Inner product space (6.7)
  • Cauchy-Schwarz inequality (6.7)
  • Symmetric matrix (7.1)
  • Quadratic form (7.2)

MATH 54 - FINAL EXAM STUDY GUIDE 5

IMPORTANT THEOREMS

Know the following theorems (or write them on your cheat-sheet)

  • Theorem 4 in section 1.4 (page 45)
  • Theorem 12 in section 1.9 (pages 90-91)
  • The invertible matrix theorem - Theorem 8 in section 2.3 and Theorem in section 2.9 and Theorem in section 4.6 (pages 131 and 165 and 255)
  • The Spanning Set Theorem - Theorem 5b in section 4.3 (page 227)
  • Theorem 12 in section 4.5 (page 247)
  • The Rank-Nullity Theorem - Theorem 14 in section 4.6 (page 253)
  • The Diagonalization Theorem - Theorem 5 in section 5.2 (page 288)
  • Theorem 6 in section 5.2 (page 291)
  • Theorem 5 in section 6.2 (page 325)
  • Theorems 6 and 7 in section 6.2 (page 330)
  • Theorem 10 in section 6.3 (page 339)
  • The Cauchy-Schwarz inequality and the Triangle inequality - Theorems 16 and 17 in section 6.7 (page 372)
  • Theorem 1 in section 7.1 (page 390)
  • Theorem 2 in section 7.1 (page 391)
  • Existence and uniqueness theorem: Theorem 1 in section 4.2 and Theorem 1 in section 6.1 (pages 457 and 538)
  • Pointwise convergence of Fourier series: Theorem 2 in section 10.3 (page 653)