Exam 3 Partial Solutions
December 1, 2011
Name:
5. (b) The hint gives a nice way of doing this problem. Another logical way
to approach this problem is to generate a basis for Rn and work down to a
basis for V W . To see a little variation, Ill give a sol
September 19, 2011 Derek Olson
1
1.1.4
a Name the two trivial subspaces of Rn .
The two trivial subspaces are Rn and cfw_0.
b Let S1 R2 be the unit circle of equation x2 + y 2 = 1. Do there exist any
two elements of S1 whose sum is an element of this set?
September 21, 2011 Derek Olson
1
1.3.4
v1
2v1
Let T be a linear transformation such that T v2 = v2 . What is the matrix for T .
v3
v3
The ith column of the matrix for T is T ei where ei is the ith standard basis vector. These
are
1
2
0
0
0
0
0 = 0 , T 1
Math 3592
October 2, 2011
Derek Olson
(Exercise 2) For each of the following subsets, determine whether it is open, closed,
both, or neither.
(a) V = cfw_(x, y, z ) R3 | z = 0.
This set is closed. We prove this by proving the complement is open. Take
(a,
Partial Homework Solutions
October 31, 2011
Derek Olson
1.8.9 Let : R R be any dierentiable function. Show that the funcction f (x, y ) =
y(x2 y 2 ) satises
1 f
1
1 f
(x, y ) +
(x, y ) = 2 f (x, y ).
x x
y y
y
(1)
Since we are dealing with a real valued f
Math 3592H
Fall 2011.
Midterm I
Name: Derek Olson
Student #
Problem
Max Possible
1
2
3
4
20
20
20
5 total
20
20
Score
1
100
1. State two equivalent denitions for a set X Rn to be closed.
A set X is closed if and only if the complement, X C , is open.
A se
Math 3592H
Fall 2011.
Midterm II
Name:
Student #
Problem
Max Possible
1
2
3
4
20
20
20
5 total
20
20
Score
1
100
1. What is the denition of the derivative of a mapping f : Rn Rm .
Let f (x) =
x2
1
x2
2
x2
3
+ x3
1
2.
. Compute the derivative of f at the p
Math 3592H
Fall 2011.
Midterm III
Name:
Student #
Problem
Max Possible
1
2
3
4
20
20
20
5 total
20
20
Score
1
100
1. Let
A=
204 2
1 1 1 1
dene a linear transformation A : R4 R2 . Find a basis for img A.
Find a basis for ker A.
2
2. What does it mean for a
Quiz 1
September 14, 2011
1. Dene a subspace of Rn .
2. Let V be a subspace of Rn . Prove that V contains the zero vector. (The
vector whose entries are all 0).
1
Quiz 2
September 22, 2011
Name:
1. Let v, w Rn .
(a) (3 Points) State Schwarzs Inequality (also called the Cauchy-Schwarz Inequality) for v
and w making sure to include the conditions under which equality holds.
(b) (2 Points) Dene what it means for v and
Quiz 3
September 29, 2011
Name:
1. (5 Points) State the denition of a convergent sequence in Rn .
2. (5 points) Prove or disprove the following statement: If cfw_an and cfw_bn
are sequences in R such that cfw_an converges and cfw_an bn converges, then
Quiz 4
October 13, 2011
Name:
1. (5 Points) State the mean value theorem. Be sure to include all
appropriate hypotheses.
2. (5 points) Prove that if f : R R is continuous on [a, b], dierentiable
on (a, b), and f (x) = 0 for all x (a, b), then f is constan
Quiz 5
October 20, 2011
Name:
1. (5 Points) Give the limit denition of the derivative of a function
F : U Rm at the point x0 U where U is an open subset of Rn .
2. (5 points) Compute the derivative of the function F : R3 R3 dened by
x
x sin z cos y
F y =
Quiz 6
October 27, 2011
Name:
1. (5 Points) Let U Rn and V Rm be open sets. State the chain rule for
two mappings g : U V and f : V Rp .
2. (5 points total) Suppose F : Rn Rm is a dierentiable function on all
of Rn . Let A : Rp Rn be a linear transformati
Quiz 7
November 2, 2011
Name:
1. (5 Points) State sucient conditions on the partial derivatives of
F : U R for F to be dierentiable on U where U is an open subset of Rn .
2. (5 points) Dene F : R2 R by
xy
2 2
x +y
x
F
=
y
0
if
x
y
=
0
0
if
x
y
=
0
0
Pro
Quiz 8
November 17, 2011
Name:
1. (5 Points) Dene the image of a linear transformation T : Rn Rm .
2. (5 points) Prove that the image of a linear transformation T : Rn Rm
is a subspace of Rm .
Quiz 9
December 7, 2011
Name:
1. (5 Points) Let U be an open subset of Rn with f : U Rn . What does it
mean for f to satisfy a Lipschitz condition?
2. (5 points) Dene f : R R by f (x) = |x|2 . Show that f satises a
Lipschitz condition (or is Lipschitz) on