260, Math Spring 2012
Jerry L. Kazdan
Problem Set 2
Due: In class Thursday, Jan. 26. Late papers will be accepted until 1:00 PM Friday.
1. Which of the following sets of vectors are bases for R2 ?
a). {(0, 1), (1, 1)}
d). {(1, 1), (1, 1)}
b). {(1, 0), (0, 1), (1, 1)}
e). {((1, 1), (2, 2)}
c). {(1, 0), (1, 0}
f). {(1, 2)}
2. a) Compute the dimension of the intersection of the following two planes in R3
x + 2y z = 0,
3x 3y + z = 0.
b) A map L : R3 R2 is dened by the matrix L :=
1
1 1
. Find the
3 3
1
nullspace (kernel) of L .
3. For which real numbers x do the vectors: (, 1, 1, 1), (1, , 1, 1), (1, 1, , 1), (1, 1, 1, )
not form a basis of R4 ? For each of the values of x that you nd, what is the dimension
of the subspace of R4 that they span?
4. Compute the dimension and nd bases for the following linear spaces.
a) Real 4 4 matrices whose diagonal elements are zero.
b) Quartic polynomials p with the property that p(2) = 0 and p(3) = 0.
c) Cubic polynomials p(x, y ) in two real variables with the properties: p(0, 0) = 0,
p(1, 0) = 0 and p(0, 1) = 0. [Suggestion: As a warm-up exercise, try quadratic
polynomials p(x, y ) = a + bx + cy + dx2 + exy + f y 2 with p(1, 0) = 0.]
d) The space of linear maps L : R5 R3 whose kernels contain (0, 2, 3, 0, 1). [Suggestion: As a warm-up exercise, try linear maps L : R3 R2 whose kernels
contain (0, 2, 3).]
5. Let C (R) be the linear space of all continuous functions from R to R .
a) Let Sc be the set of dierentiable functions u(x) that satisfy the dierential equation
u = 2xu + c
for all real x . For which value(s) of the real constant c is this set a linear subspace
of C (R)?
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b) Let C 2 (R) be the linear space of all functions from R to R that have two continuous
derivatives and let Sf be the set of solutions u(x) C 2 (R) of the dierential
equation
u + 2xu = f (x)
for all real x . For which polynomials f (x) is the set Sf a linear subspace of C You (R)?
[Note: are not being asked to solve this equation.]
c) Let A and B be linear spaces and L : A B be a linear map. For which vectors
y B is the set
Sy := {x A | Lx = y }
a linear space?
6. Let U and V both be two-dimensional subspaces of R5 .
a) Let W := U V . Show that W is a linear space and nd all possible values for
the dimension of W .
b) Let Z := U + V as the set of all vectors z = u + v where u U and v V can
be any vectors. Show that W is a linear space and nd all possible values for the
dimension of W .
7. Linear maps F (X ) = AX , where A is a matrix, have the property that F (0) = A0 = 0,
so they necessarily leave the origin xed. It is simple to extend this to include a
translation,
F (X ) = V + AX,
where V is a vector. Note that F (0) = V . [These are called ane maps].
a) Find the vector V and the matrix A that describe each of the following mappings
[here the light blue F is mapped to the dark red F ].
ii).
i).
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iii).
iv).
b) Show that the set of all ane maps from
its dimension?
R2
to R2 forms a linear space. What is
8. a) Find set Q of all linear maps L : R3 R3 whose nullspace is exactly the plane
{ (x1 , x2 , x3 ) R3 | x1 + 2x2 x3 = 0 } .
b) Is this set a linear space? If so, what is its dimension?
Bonus Problem
[Please give these directly to Professor Kazdan]
1-B In the notes on Symmetries:
http://www.math.upenn.edu/ kazdan/260S12/notes/symmetries.pdf
do the Exercise at the bottom of page 2:
a) Use RS = SR3 to show that the maps RSR , R2 S , and RSR1 are in the list
(5).
b) Prove that the list (5) really does contain all the symmetries of the square. I
suggest beginning with the special case where the vertex A is xed. What are the
possible adjacent vertices? A key ingredient is that the symmetries of the square
are rigid motions, that is, they preserve distances between points, so no stretching
or shrinking is allowed.
[Last revised: January 24, 2012]
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