Mathematics 54W Professor A. OgusFall 2007Practice FinalWork each problem on a separate sheet of paper. Be sure to put your name,your section number, and your GSI’s name on each sheet of paper. Also,at the top of the page, in the center, write the problem numbe r, and besure to put the pages in order. Write clearly—explanations (with completesentences when appropriate) will help us understand what you are doing.Math 49 students taking the linear algebra portion should work problems1–5; Math 49 students taking the differential equation should work problems6–10.1. Let A :=1 −2 1 1 −2−1 2 −1 0 11 −2 1 −1 0.(a) Find a matrix B which is in reduced row echelon form and whichis row equivalent to A.(b) Find a basis for the null space of A.(c) Find a basis for the column space of A, from among the columnsof A.(d) Find all X such that AX =2−10.(e) Find at least one X such that ATAX = ATB, where B =3−21.Hint: you do not need to calculate ATA to do this.2. Write the definition of each of the following concepts. Use completesentences and be as precise as you can.(a) The inverse of a matrix.(b) A linearly independent sequence in a vector space V .(c) The dimension of a vector space. State the theorems w hich makesthis definition meaningful.(d) The orthogona l projection of a vector in Rnonto a linear subspaceW .(e) An eigenvector of a linear operator T : V → V .3. Let V be the space of vectors in R4such that x1+ x2+ x3+ x4= 0.and let W be the set of vectors in V such that x1= x4.(a) Find an orthogonal basis (v1, v2, v3) for V with v1= (0, 1, −1, 0)and such that (v1, v2) is a basis for W .(b) Find the orthogonal projection of v := (2, 1, 3, −6) on W .(c) Find the distance from v to W .4. Suppose that A is a matrix with 4 rows and 8 columns, and supposethat the rows of A span a three dimensional subspace of R8. Answerthe following questions, explaining you reasoning.(a) What is the dimension of the space spanned by the columns ofA?(b) What is the dimension of the null space of A?(c) What is the dimension of the null space of ATA?(d) Prove that for any m × n real matrix A, the ranks of ATA andof A are the same.5. Let A =6 −14 2.(a) Find the eigenvalues of A.(b) We know that there exist matrices S and T such that A = ST S−1,where T is upper triangular. Find T.(c) Now com pute etA, as a function of t.(d) Find a matrix B with positive eigenvalues such that B2= A.6. Consider the system of differential equations:f0= gg0= 2f − g(a) Write this system as a vector-valued differential equation.(b) Find a fundamental solution set for the equation in part (a).(c) Compute the Wronskian of your solution set.(d) Find a pair of functions f, g satisfying the original system andsuch that f(0) = −1 and g(0) = 5.7. For each of the following matrices, sketch and describe the trajectoriesof the solutions to the differential equation Y0(t) = AY (t). In par-ticular, exhibit any asymptotes and/or invariant lines, draw arrowsindicating the direction of the flow along the solution, etc.(a) A =2 00 1(b) A =1 00 −2(c) A =1 −22 0(d) A =−1 10 128. Suppose Y00(x) = kY (x) and Y (0) = 0, Y0(3) = 0.(a) For which values of k ∈ R is there a nontrivial solution to thisequation?(b) Explain (prove) why you have found all such k. (This part willcount more than the ac tual answer.)(c) For each such k, give the corresponding solutions.9. Suppose f: R → R is even, periodic with period 2π, and satisfiesf(x) = sin x for x ∈ [0, π].(a) Draw a sketch of the graph of f , labeling your axes carefully.(b) Find a Fourier series which represents f. Explain how you findthe coefficients. You may use one or more of the formulas at theend of the test to evaluate the coefficients if you like.(c) Does the series converge to f at every point? Explain.10. Suppose that a bar of length π with thermal coefficient α2= 2 isinsulated on its surface except at the end points.(a) Write the differential equation governing heat diffusion in the bardescribed above.(b) If the ends of the bar are kept at 0oand the initial temperaturedistribution is u(x, 0) = sin(5x), find a formula for the tempera-ture distribution at all times.(c) If instead one end is kept at π◦and the other at 3π◦, what is thelimiting temperature distribution (steady state solution) of thetemperature as t → ∞? Verify directly that this limit distributionsatisfies the equation in part (a).(d) In the situation (c), assume again that the initial temperature isgiven by u(x, 0) = sin(5x). Find a formula for the temperaturedistribution at time t. You may use one or more of the formulaslisted at the end of the test.3Formulas1 = 2X1 + (−1)k+1πksin(kx) for 0 ≤ x ≤ πx = 2X(−1)k+1ksin(kx) for 0 ≤ x ≤ πsin(nx) cos(x) =sin((1 + n)x) + sin((n − 1)x)2cos(nx) sin(x) =sin((1 + n)x) + sin((1 −
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