MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #1, Week 2 (Dot and Cross Products, Euclidean Geometry of Rn )
Authors: Paul Bamberg and Kate Penner
R scripts by Paul Bamberg
Last modied: September 22, 2015 by Paul Bamberg (in p
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 1
Problem 1: In this problem we give a mathematical foundation to the decimal expansion of real numbers. Let x be a positive real number. We dene
an innite sequence of integers (
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 5
This Problem Set will be entirely devoted to complete the proof of the equivalence
between the two denitions of compact sets, the one in terms of limit points and
the one in te
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 6
Problem 1: Consider the function : R R3 given by
(t) = (a cos t, a sin t, bt) .
(a) The graph of this function denes a curve in R4 . Find the tangent
line L to at t = 2.
(b) Pr
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 7
Problem 1: A function f : Rn R is said to be homogeneous if f (tx) =
tf (x) , t R, x Rn .
(a) Prove that, if f : Rn R is dierentiable and homogeneous, then it
must be such that
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 11
Problem 1: Let f, g : Rn R be two function bounded, with bounded
support and integrable.
(a) Prove that f 2 is integrable.
(b) Prove that f g is integrable.
(Hint: (a) We may
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 8
Problem 1: Let f (x, y, z) = xy 2 z 3 , and consider the point a = (1, 0, 1).
(a) Find the second-order Taylor polynomial P2 (h) of f (x) at x = a.
(b) Show directly (i.e. by c
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 9
Problem 1: Use the method of Lagrange multipliers to nd the points on
the line x + y = 10 and the ellipse x2 + 2y 2 = 1 which are closest.
Problem 2: Find the points in the clo
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 12
Problem 1: Compute 2xyzdx + x2 zdy + x2 ydz, where is a smooth curve
in R3 starting at (1, 1, 1) end ending at (1, 2, 4).
Problem 2: Use the denition of surface integral to pr
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 10
Problem 1: Suppose f, g : Rn R are admissible functions, and let c R.
Prove that f + g and cf are admissible. Namely the set of admissible
functions is a vector space.
Problem
Mathematics 23b, Spring 2006
Theoretical Linear Algebra and
Multivariable Calculus II
Catalog Number: 8571
Exam Group: 4
COURSE INFORMATION
Instructor: Alberto De Sole
contact info: Sc. Ctr. #331 , 617-496-5211 , desole@math.harvard.edu
Course Assistant
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 4
Problem 1: Let X be a metric space, a X, and let f be a function f :
X R.
Denition 0.1. We say that the limit of f for x going to a is (plus or
minus) innity, denoted
lim f (x)
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 3
Problem 1: Let (x1 , x2 , . . . ) be a sequence in a metric space X. Prove or
disprove the following statements:
(1) if there is a unique limit point x for the sequence, then t
MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #3, Week 1
Author: Paul Bamberg
R scripts by Paul Bamberg
Last modied: November 5, 2015 by Paul Bamberg (improved wording of proof
9.2)
Reading
Hubbard, Section 1.5. The only topo
MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #3, Week 2
Author: Paul Bamberg
R scripts by Paul Bamberg
Last modied: July 27, 2015 by Paul Bamberg
The lecture outline and problems have not yet been revised for 2015. Pages
1-6
MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #3, Week 3
Dierentiability, Newtons method, inverse functions
Author: Paul Bamberg
R scripts by Paul Bamberg
Last modied: July 26, 2015 by Paul Bamberg
The lecture outline and prob
MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #3, Week 4
Implicit functions, manifolds, tangent spaces, critical points
Author: Paul Bamberg
R scripts by Paul Bamberg
Last modied: July 27, 2015 by Paul Bamberg
The lecture outl
Math 23b: Theoretical Linear Algebra
and Multivariable Calculus I
Practice questions for the nal exam
May 12, 2006
Problem 1
Decide whether the following statements are True or False. (Note: There is no need
to justify your answers, just circle T or F. Yo
Math 23b: Theoretical Linear Algebra
and Multivariable Calculus II
MIDTERM EXAM 2
April 17, 2006
Your name:
Problem
1
2
3
4
5
Total
Points
21
20
20
20
20
101
Score
In the following problems you can use any of the results we have
proved in class, if you st
Math 23b: Theoretical Linear Algebra
and Multivariable Calculus II
MIDTERM EXAM 1
March 6, 2006
Your name:
Problem
1
2
3
4
5
Total
Points
20
20
20
20
20
100
Score
In the following problems you can use any of the results we have
proved in class, if you sta
Math 23b - Practice Midterm Exam 1
In the following problems you can use any of the results we have proved in class, if
you state them clearly before using them.
Problem 1: Prove or disprove each of the following statements:
(a) Every set in Rn is either
Math 23bTheoretical Linear Algebra
and Multivariable Calculus II
PROBLEM SET 2
Problem 1: Prove that the equinumerosity between sets, A B, is an equivalence concept (we cannot talk of equivalence relation, since its not dened in a set, but in the family o
Mathematics 23b, Spring 2006
Theoretical Linear Algebra and
Multivariable Calculus II
Catalog Number: 8571
Exam Group: 4
COURSE INFORMATION
Instructor: Alberto De Sole
contact info: Sc. Ctr. #331 , 617-496-5211 , desole@math.harvard.edu
Course Assistant
Mathematics 23b, Spring 2006
Theoretical Linear Algebra and
Multivariable Calculus II
Catalog Number: 8571
Exam Group: 4
COURSE INFORMATION
Instructor: Alberto De Sole
contact info: Sc. Ctr. #331 , 617-496-5211 , desole@math.harvard.edu
Course Assistant
Math 23aTheoretical Linear Algebra
and Multivariable Calculus I
PROBLEM SET 3
Problem 1: (a) Let V be a given vector space over a certain eld F. Prove
that (1) v = v
x
(b) Prove that the subset U R3 consisting of all 3-columns y R3
z
such that 2x + y = 0
Math 23aTheoretical Linear Algebra
and Multivariable Calculus I
PROBLEM SET 1
Problem 1: Consider the following four sets:
A = cfw_1, 2 , B = cfw_1, cfw_2 ,
C = cfw_1, cfw_1, 2 , D = cfw_1, cfw_2, cfw_1, 2 .
For each of the following statements, argue whe
Math 23aTheoretical Linear Algebra
and Multivariable Calculus I
PROBLEM SET 4
Problem 1: Suppose W is a vector space over some eld F, and let U, V be
two subspaces of W . In the exam you (were supposed to) prove that U V
is again a subspace of W , while i
Math 23aTheoretical Linear Algebra
and Multivariable Calculus I
PROBLEM SET 5
Problem 1: Let A be an m n matrix with real coecients, i.e.
11 12 1n
22 2n
A = 21
m1 m2 mn
Recall that the columns of A are, by denition, the m-column vectors,
11
12
1n
Math 23aTheoretical Linear Algebra
and Multivariable Calculus I
PROBLEM SET 6
Problem 1: Recall we denote by Pn the vector space of all polynomials in
x with real coecients of degree less than or equal to n. Consider the
following linear functions:
x
I :
Math 23aTheoretical Linear Algebra
and Multivariable Calculus I
PROBLEM SET 9
Problem 1: A) Recall we dened in class the complex conjugate of a complex
number z = + i to be z = i. In this (and the next) problem,
consider R as a subset of C by thinking at
Single and Multivariable
Calculus
Early Transcendentals
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