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**Unformatted text preview: **Last Name: First Name: Mathematics 21b Final Examination
January, 2000 Your Section (circle one): John Boller Noam Elkies
MWF 10 MWF 11 N0 calculators are allowed. 1. [6 points] A matrix M is of the form 0 * 5 a:
>1: >I< :1: a: ’
where as usual the *’s denote unknown and possibly different real numbers. Given that M is in row-reduced echelon form, ﬁnd all possible M, and explain Why there are no other
possibilities. For each of the M that you have found, determine its rank, image, and kernel. 2. Consider the matrix A: r—li—lw
l—‘NI—i
MI—‘l—d . a) [4 points] Construct an orthonormal eigenbasis for A. b) [2 points] Is A diagonalizable? Why or why not? c) [4 points] Using parts (a) and (b), compute A2000. 3. Let B be the matrix 1.2 —0.4 0 0
1.3 0.4 0 0
O 0 1.2 0.4
0 O —2.6 0.8 a) [3 points] Find all eigenvalues of B. b) [3 points] Does the dynamical system :29 + 1) = 3m) - have a point of stable equilibrium? Why or why not? c) {4 points] Describe qualitatively the behavior of this dynamical system if 18(0) is the unit
vector ' - 0
1
0
D 4. Consider the linear transformation T : C°° —> C°° given by Tm = f” — 2 f’. a) [6 points] Find all real eigenvalues of T and their corresponding eigenspaces. b) [2 points] Let T2 be the same linear transformation restricted to the subspace P2 of CC”,
consisting of polynomials of degree at most 2. (That is, T2 : P2 —> P2 is the transformation
taking any polynomial f of degree at most 2 to f" - 2}“ Choose a basis for P2, and write the matrix A2 of T2 with respect to this basis. c) [4 points] Find the image and kernel of this matrix A2. Check (part of) your work by
explaining the relationship between parts (a) and ' 833' a? —- (Uhllﬂ? 8y _ a *- ($+y~3)y_ We consider only the ﬁrst quadrant m 2 0, y 2 0 since we are modelling populations. a) [2 points] Sketch the nullclines of this system, and ﬁnd any equilibrium points. b) [4 points] Use linear approximation to ﬁnd the Jacobian J of the system at any equilibrium
points from part (a). C) [4 points] Analyze the stability of each equilibrium point by computing the eigenvalues of J. d) [2 points] Sketch (approximately) the phase plane of this system, including behavior near equilibrium points and approximate direction of the ﬂow lines in the regions separated by the
nullclines. '10 6. Consider the temperature T(m,t) of a metal bar extending from m : 0 to a: = 71'. The
temperature satisﬁes the heat equation ' 8T _ BZT
8t _ ‘”’ 8:32
and the ends are held at a constant temperature of zero, i.e. T(O,t) = O and T(7r,t) z 0 for allth. a) [3 points] Show that the functions e”"“”2‘sin(nx) satisfy the differential equation and the
initial conditions for all positive integers n. 11 b) [7 points] Suppose now that the initial temperature of the bar is given by
T(sc, 0) = 6(5):) = sin3 3:.
Determine T(a:,t) for all a: E [0,7r] and all times t 2 0. [Hintz Use Euler’s formula ell” = cosy + ésiny if you are not sure about the relevant trigonometric identities.] Examine the
behavior of T as t —> oo. 12 ...

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