260, Math Spring 2012
Jerry L. Kazdan
Problem Set 3
Due: In class Thursday, Feb. 2. Late papers will be accepted until 1:00 PM Friday.
1. Say you have k linear algebraic equations in n variables; in matrix form we write
AX = Y . Give a proof or counterexample for each of the following.
a) If n = k there is always at most one solution.
b) If n > k you can always solve AX = Y .
c) If n > k the nullspace of A has dimension greater than zero.
d) If n < k then for some Y there is no solution of AX = Y .
e) If n < k the only solution of AX = 0 is X = 0.
2. Let A and B be n n matrices with AB = 0. Give a proof or counterexample for
each of the following.
a) BA = 0
b) Either A = 0 or B = 0 (or both).
c) If B is invertible then A = 0.
d) There is a vector V = 0 such that BAV = 0.
3. Consider the system of equations
x+y z = a
x y + 2z = b.
a) Find the general solution of the homogeneous equation.
b) A particular solution of the inhomogeneous equations when a = 1 and b = 2
is x = 1, y = 1, z = 1. Find the most general solution of the inhomogeneous
equations.
c) Find some particular solution of the inhomogeneous equations when a = 1 and
b = 2.
d) Find some particular solution of the inhomogeneous equations when a = 3 and
b = 6.
[Remark: After you have done part a), it is possible immediately to write the solutions
to the remaining parts.]
4. Let A =
1
1 1
1 1
2
.
a) Find the general solution Z of the homogeneous equation AZ = 0.
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b) Find some solution of AX =
1
2
c) Find the general solution of the equation in part b).
d) Find some solution of AX =
1
2
e) Find some solution of AX =
3
0
f ) Find some solution of AX =
7
. [Note: ( 7 ) = ( 1 ) + 2 ( 3 )].
2
2
0
2
and of AX =
3
6
[Remark: After you have done parts a), b) and e), it is possible immediately to write
the solutions to the remaining parts.]
5. Let A : R3 R2 and B : R2 R3 , so BA : R3 R3 and AB : R2 R2 .
a) Show that BA can not be invertible.
b) Give an example showing that AB might be invertible.
6. Given the ve data points:
P1 = (1, 1),
P2 = (0, 0),
P3 = (1, 0),
P4 = (2, 2),
P5 = (3, 0),
nd the (unique!) quartic polynomial p(x) that passes through these points. [Dont
bother to simplify your answer.]
The next sequence of problems involve techniques for explicitely solving the ordinary dierential equation
Lu := a(x)u + b(x)u + c(x)u = f (x)
in the very special (but important) special case where the coecients a(x) , b(x) , and c(x)
are constants with a = 0 , and the right hand side f (x) is simple. These assume you have
mastered the ideas concerning the complex exponential ex+iy in
http://www.math.upenn.edu/ kazdan/260S12/hw/hw0.pdf
7. Homogeneous equation example
a) Let Lu := u + u 2u . Find two linearly independent solutions of the homogeneous
equation of the form u(x) = erx where r is a constant, possibly complex.
b) Seek (and nd) solutions u(t) of u + 2u + 5u = 0 in the form u(t) = erx , where
r might be a complex number. Use this to nd two linearly independent real
solutions. [See Homework Set 0].
c) Find a solution of u + 2u + 5u = 0 that satises the initial conditions u(0) = 1,
u (0) = 0.
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8. Homogeneous Let equation Lu := au + bu + cu where the coecients are real constants with a = 0. Show that L(erx ) = p(r)erx , where p(r) is a quadratic polynomial.
Assume the roots of p(r) = 0 are distinct
a) Find two linearly independent solutions (possibly complex) of the homogeneous
equation Lu = 0.
b) If the solutions you just found are complex-valued functions, use them to nd two
linearly independent real solutions.
9. Inhomogeneous equation Example: Find some particular solution v (x) of v +
v = x2 1. [Suggestion: Since the right hand side is a quadratic polynimial and
the coecients of the dierential equation are constants, seek v (x) as a quadratic
polynomial: v (x) = A + Bx + Cx2 ].
This experiment leads one to the general approach of the next problem on nding a
particular solution of the inhomogeneous equation when f (x) is a polynomial.
10. Inhomogeneous equation: polynomial Let PN be the linear space of polynomials
of degree at most N and L : PN PN the linear map dened by Lu := au + bu + cu ,
where a , b , and c are constants. Assume c = 0 (and a = 0).
a) Compute L(xk ).
b) Show that nullspace (=kernel) of L : PN PN is 0. [A strict proof uses induction
but it is convincing enough to treat the case N = 3.]
c) Show that for every polynomial q (x) PN there is one and only one solution
p(x) PN of the ODE Lp = q . [A strict proof uses induction but it is convincing
enough to treat the case N = 3.]
11. Inhomogeneous equation Example: Use the observation in Problem 8 to nd
particular solutions of
a) u 4u = 2e3x
b) u 4u = cos x [Hint: cos x is the real part of eix .]
c) u 4u = cos x + 2 sin x
d) u 4u = ex cos x . [Not assigned but useful.]
Bonus Problems
[Please give these directly to Professor Kazdan]
1-B [Error in Interpolation] Let x0 < x1 < x2 be distinct real numbers and f (x)
a smooth function. In class we showed there is a unique quadratic polynomial p(x)
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with the property that p(xj ) = f (xj ) for j = 0, 1, 2. Here you nd a formula for the
error:= f (x) p(x).
If b is in the open interval (x0 , x2 ) with b = xj , j = 0, 1, 2, show there is a point c
(depending on b ) in the interval (x0 , x2 ) so that
f (b) = p(b) +
f (c)
(b x0 )(b x1 )(b x2 ).
3!
This estimate is related to the procedure used to nd the remainder in a Taylor polynomial.
[Suggestion: Dene the constant M by
f (b) = p(b) + M (b x0 )(b x1 )(b x2 ),
and look at
g (x) := f (x) [p(x) + M (x x0 )(x x1 )(x x2 )].
Now observe that g (x) = 0 at x0 , x1 , x2 , and b (by denition of M ).]
2-B Let Lu := u + bu + cu = 0, where b and c are constants.
a) If w(x) is a solution of the homogeneous equation Lw = 0 with initial conditions
w(0) = 0 and w (0) = 0, show that w(x) = 0 for all x 0.
b) Make the change of variable x = t and show that as a function of t w satises:
dw
d2 w
b
+ cw = 0
dt
dt
with
w(0) = 0
and w (0) = 0.
This has the same structure as the original equation, only the sign of b is ipped
so by applying part a), conclude that w(x) = 0 for all x .
c) Use this to state and prove a uniqueness theorem for the inhomoheneous equation
Lu = f (x) with u(0) = and u (0) =
[Last revised: February 9, 2012]
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