Math 21b final exam guide

 

Regular sections

 

 

a)  The exam

 

∑       The final exam is on Friday, May 21 from 9am to 12:15pm.  Go to Harvard Hall 202 if the first letter of your last name is between A-N inclusive,  If the first letter of your last name is between O-Z inclusive, go to Harvard Hall 201.

 

∑       Here is an approximate facsimile of the first page of the exam booklet:

 

 

MATH 21b Final Exam

S 2004

Regular sections

               

 

 

Name:  _____________________________________________

 

Circle the time of your section: 

 

MWF10   ·   MWF11   ·   MWF12   ·   TTH10

 

Instructions: 

·         This exam booklet is only for students in the Regular sections.

·         Print your name in the line above and circle the time of your section.

·         Answer each of the questions below in the space provided. If more space is needed,

use the back of the facing page or on the extra blank pages at the end of this booklet.  Please direct the grader to the extra pages used.

·         Please give justification for answers if you are not told otherwise.

·         Please write neatly.  Answers deemed illegible by the grader will not receive credit.

·         No calculators, computers or other electronic aids are allowed; nor are you allowed to

refer to any written notes or source material; nor are you allowed to communicate with other students.  Use only your brain and a pencil.

·         Each of the problems counts for the same total number of points, so budget your 

      time for each problem.

∑        Do not detach pages from this exam booklet.

 

 

 

In agreeing to take this exam, you are implicitly agreeing to act with fairness and honesty.

 

 

∑       There will probably be a mix of true/false problems, multiple choice problems, and problems of the sort that you worked for the homework assignments. 

∑       The exam will cover the material in Chapters 1-3, 5, 6.1, 6.2, 7, 8.1, 9.1 and 9.2 in the text book, Linear Algebra and Applications.  The exam will also cover the material in Otto Bretscher’s handout on non-linear systems and the material in the Differential Equation handout.

∑       Advice for studying:  There are plenty of answered problems in the text book and I strongly suggest that you work as many of these as you think necessary.  I have supplied in a separate handout some problems (with answers) to test your facility with the material in the Differential Equations handout.     

∑       Old exams:  Old exams will not be terribly useful for the three reasons.  The exams from previous semesters will either test on material that we have not taught, or not test material that we have.  Second, this version of Math 21b is not the same as those taught in previous semesters.  Finally, the exam format used in previous years might not be the format we will use.  In any event, some old exams are archived in Cabot Library.

∑       Of the topics covered, some are more important than others.  Given below is a list to guide your review towards the more central issues.

 

 

b)  Topics and issues to focus on.

 

∑       Be able to find the matrix that corresponds to a linear system of equations.

∑       Be able to find rref(A) given the matrix A.

∑       Be able to solve A =  by computing rref for the augmented matrix, thus rref(A|).

∑       Be able to find the inverse of a square matrix A by doing rref(A|I) where I is the identity matrix.

∑       Be able to use rref(A) to determine whether A is invertible, or if not, what its kernel is and what its image dimension is.

∑       Become comfortable with the notions that underlie the formal definitions of the following terms: linear transformation, linear subspace, the span of a set of vectors, linear dependence and linear independence, invertibility, orthogonality, kernel, image.

∑       Given a set, {1, . . . , k}, of vectors in Rn, be able to use the rref of the n-row/k-column matrix whose j’th column is j to determine if this set is linearly independent.

∑       Be able to find a basis for the kernel of a linear transformation.

∑       Be able to find a basis for the image of linear transformation.

∑       Know how to multiply matrices and also matrices against vectors.  Know how these concepts respectively relate to the composition of two linear transformations and the action of a linear transformation.

∑       Know how the kernel and image of the product, AB, of matrices A and B are related to those of A and B.

∑       Understand how rref(AB) relates to rref(A) and rref(B).

∑       Be able to find the coordinates of a vector with respect to any given basis of Rn.

∑       Be able to find the matrix of a linear transformation of Rn with respect to any given basis.

∑       Understand the relations between the triangle inequality ( |+| ≤ || + ||), the Cauchy-Schwarz inequality (|·| ≤ ||||), and the Pythagorean equality

(|+|2 = ||2 + ||2).

∑       Be able to provide an orthonormal basis for a given linear subspace of Rn.  Thus, understand how to use the Gram-Schmidt procedure.

∑       Be able to give a matrix for the orthogonal projection of Rn onto any given linear subspace.

∑       Be able to work with the orthogonal complement of any given linear subspace in Rn.

∑       Recognize that rotations are orthogonal transformations. 

∑       Be able to recognize an orthogonal transformation:  It preserves lengths.  Such is the case if and only if its matrix, A, has the property that  |A| = || for all vectors .  Equivalent conditions:  A-1 = AT.  Also, the columns of A form an orthonormal basis.  Also, the rows of A form an orthonormal basis. 

∑       Remember that the transpose of an orthogonal matrix is orthogonal, as is the product of any two orthogonal matrices.

∑       Be able to recognize symmetric and skew-symmetric matrices.

∑       Recognize that the dot product is matrix product, · = xTy, where x and y on the right hand side of the inequality are respectively viewed as an n ´ 1 and 1 ´ n matrix.

∑       Recognize that kernel(A) = kernel(ATA).

∑       Be able to find the least square solution of A =  is * º (ATA)-1AT.

∑       Be able to use least squares for data fitting:  Know how to find the best degree n polynomial that fits a collection {(xk, yk)}1≤k≤N of data points.

∑       Know how to compute the angle between two vectors from their length and dot product:  cos(q) = ·/(|| ||).

∑       Know how to compute the determinant of a square matrix.

∑       Know the properties of the determinant:  det(AB) = det(A)det(B), det(AT) = det(A), det(A-1) = 1/det(A), det(SAS-1) = det(A).

∑       Know how the determinant is affected when rows are switched, or columns are switched, or when a multiple of one row is added to another, or a multiple of one column is added to another.

∑       Know that det(A) = 0 if and only if kernel(A) has dimension bigger than 1.

∑       Know that trace(A) = A11+A22+···+Ann.

∑       Know the characteristic polynomial, l ® Ã(l) = det(lI – A) and understand its significance:  if Ã(l) = 0, there is a non-zero vector  such that A = l.

∑       Realize that Ã(l) factors completely if one allows complex roots.

∑       Be able to comfortably use complex numbers.  Thus, multiply them, add them, use the polar form a+ib = reiq = r cosq + i r sinq. 

∑       Understand that the norm of |a+ib| is (a2 + b2)1/2 and be comfortable with the operation of complex conjugation that changes z = a+ib to  = a-ib.  In this regard, don’t forget that |z| = || 

∑       Be comfortable with the fact that |zw| = |z| |w| and that |z+w| ≤ |z| + |w| and that these hold for any two complex numbers z and w.

∑       Understand that if l is a root of Ã, then so is its complex conjugate.

∑       Know what an eigenvalue, eigenvector and an eigenspace are.

∑       Understand the difference between the algebraic multiplicity of a root of the characteristic polynomial and its geometric multiplicity as an eigenvalue of A. 

∑       Understand that the kernel of A-lI is the eigenspace for the eigenvalue l.

∑       Understand that if A has an eigenvalue with non-zero imaginary part, then some of the entries of any corresponding non-zero eigenvector must have non-zero imaginary part as well.

∑       Recognize that a set of eigenvectors whose eigenvalues are distinct must be linearly independent.

∑       Be able to compute the powers of a matrix a diagonalizable matrix.

∑       Know the formula for the determinant of A as the product of its eigenvalues, and that of the trace of A as the sum of its eigenvalues.

∑       Know that a linear dynamical system has the form (t+1) = A(t) where A is a square matrix.  Know how to solve for (t) in terms of (0) in the case that A is diagonalizable.

∑       Recognize that the origin is a stable solution of (t+1) = A(t) if and only if the norm of each of A’s eigenvalues has absolute value that is strictly less than 1.

∑       Be able to solve for the form of t ® (t) in terms of (0) when  = A and A is diagonalizable.

∑       Know that the solution to  = A where (t) is zero for all time is stable if and only if the real part of each eigenvalue of A is strictly less than zero.

∑       Know the definition of an equilibrium point for a non-linear dynamical system and the criteria for it’s stability in terms of the matrix of partial derivative.

∑       Know how to plot the null-clines and approximate trajectories for a non-linear dynamical system on R2.

∑       Be comfortable with viewing the set of continuous or differentiable functions on some part of the R as a linear space.  Thus, understand how the terms ‘subspace’, ‘linear dependence and independence’, ‘span’, ‘linear transformation’, ‘kernel’, ‘image’, ‘basis’, ‘dimension’ are used in this context.

∑       Understand how to view ()nf(t) + a1(t) ()n-1f(t) +···+ an(t) f(t) as the affect of a linear operator acting on the given function t ® f(t). 

∑       Understand how to view the equation ()nf(t) + a1(t) ()n-1f(t) +···+ an(t) f(t) = 0 as the criteria for f’s membership in the kernel of this operator, and how to view the existence of a solution to the equation ()nf(t) + a1(t) ()n-1f(t) +···+ an(t) f(t) = g(t) as signifying that g(t) is in the image of the operator.

∑       Be able to write the solutions to the equation ()nf(t) + a1 ()n-1f(t) +···+ an f(t) = 0 in the case that {a1, …, an} are constants using the roots of the polynomial function that sends l ® Ã(l) = ln + a1ln-1 +···+ a0.  Be aware that there is an n-dimensional space of solutions to the equation.

∑       Pay special attention to the case where à here has complex roots, or roots with algebraic multiplicity greater than 1.

∑       Be comfortable that the assignment of áf, gñ º f(t)g(t) dt  to continuous functions

f and g on the interval [-π, π] can be viewed as defining an inner product on the space of such functions. 

∑       Understand how the square of the length, || f ||2 º áf, fñ =  f(t)2 dt, can be small even if f is large at some points in [-π, π]. 

∑       Be comfortable with the fact that || f+g || ≤ || f || + || g || and that |áf,gñ| ≤ || f ||·|| g ||.

∑       Understand the notion of an orthonormal basis in the context that uses || f || for length and áf,gñ for the inner product.  Understand how to compute orthogonal projections to a subspace that has a given orthonormal basis.

∑       Understand the sense in which the constant function  with {cos(nt), sin(nt)}n=1,2,… define an orthonormal basis for the space of continuous functions on [-π, π]. 

∑       Be able to write down the integrals that give the coefficients for the expansion of any given function t ® f(t) for this basis.  (This is the Fourier expansion of f.)

∑       Understand the sense in which the truncated Fourier expansion for f that uses only N terms converges to f as N ® ∞.

∑       Understand how Fourier expansions are used to write down solutions to the heat equation, the Laplace equation and the wave equation.