Problem Set 1
MATH-23a/MATH-E23a, Fall 2012
due 9/12/2012, in class
(1) Do the following exercises in Hubbard: 0.2.1 (for each of the three statements, think
about whether the negation is true or false - you do not need to turn anything in for
this part),

Problem Set 12
MATH-23a/MATH-E23a, Fall 2012
due 12/5/2012, by 5PM in CAs mailboxes
(1) Show that the system of equations
3x + y z + u2 = 0
x y + 2z + u = 0
2x + 2y 3z + 2u = 0
can be solved for:
(a) x, y, and u in terms of z,
(b) x, z, and u in terms of

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MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #2, Week 2 (Series, Convergence, Power Series)
Authors: Paul Bamberg and Kate Penner (based on their course MATH S-322)
R scripts by Paul Bamberg
Last modied: October 15, 2015 by P

MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #1, Week 3 (Row Reduction, Independence, Basis)
Authors: Paul Bamberg and Kate Penner
R scripts by Paul Bamberg
Last modied: September 29, 2015 by Paul Bamberg (xed tiny typo in pr

MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #2, Week 4 (Derivatives, Inverse functions, Taylor series)
Authors: Paul Bamberg and Kate Penner (based on their course MATH S-322)
R scripts by Paul Bamberg
Last modied: October 2

MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #1, Week 1 (Fields, Vectors, and Matrices)
Authors: Paul Bamberg and Kate Penner
R scripts by Paul Bamberg
Last modied: September 3, 2015 by Kate Penner
Reading
Hubbard, Sections

MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #2, Week 3 (Limits and continuity of functions)
Authors: Paul Bamberg and Kate Penner (based on their course MATH S-322)
R scripts by Paul Bamberg
Last modied: October 20, 2015 by

MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #2, Week 1 (Number Systems and Sequences)
Authors: Paul Bamberg and Kate Penner (based on their course MATH S-322)
R scripts by Paul Bamberg
Last modied: October 8, 2015 by Paul Ba

Problem Set 11
MATH-23a/MATH-E23a, Fall 2012
due 11/19/2012, by 5PM in CAs mailboxes
(1) Do the following exercises in Hubbard: 2.7.1, 2.7.2
(2) Do the following exercises in Hubbard: 2.7.3, 2.7.6
(3) Let T : R3 R2 be a linear transformation dened by
x

Problem Set 10
MATH-23a/MATH-E23a, Fall 2012
due 11/14/2012, in class
Here are some important things to remember from class on Wednesday, 11/7.
Fun Fact 1. Let V Rn be a subspace. If V = cfw_0, then V has a basis.
How do we go about constructing one? Sinc

Problem Set 3
MATH-23a/MATH-E23a, Fall 2012
due 9/26/2012, in class
(1) More subspace practice: let U, V Rn be subspaces.
(a) Prove that U V is a subspace.
(b) Prove that U + V := cfw_u + v : u U, v V is a subspace.
(c) Find necessary and sucient conditi

Problem Set 2
MATH-23a/MATH-E23a, Fall 2012
due 9/19/2012, in class
(1) Some FUNction questions rst. just think about these - you do not need to turn anything
in for any of them, but you should think about them (over breakfast, or at section, or with
frie

Problem Set 4
MATH-23a/MATH-E23a, Fall 2012
due 10/3/2012, in class
(1) Do the following exercises in Hubbard: 1.5.1, 1.5.2
(2) Do the following exercises in Hubbard: 1.5.4, 1.5.6
(3) Prove that the union of any collection of open sets of Rn is an open se

Problem Set 5
MATH-23a/MATH-E23a, Fall 2012
due 10/10/2012, in class
(1) Let cfw_a1 , a2 , . . . Rn be a sequence. Show that the sequence converges to a point a Rn if and
only if every subsequence bm := ai(m) converges to a.
(2) Prove this: Let u : R R be

Problem Set 6
MATH-23a/MATH-E23a, Fall 2012
due 10/17/2012, in class
(1) Consider the construction of the Cantor Set C R from class. Call E0 the starting interval
[0, 1]. After one step, we get E1 which consists of two intervals.
(a) How many intervals ar

Problem Set 7
MATH-23a/MATH-E23a, Fall 2012
due 10/24/2012, in class
(1) How do certain sets behave under continuous maps?
(a) Let A Rn be a nonempty open set. Let f : A Rm be a continuous map. Prove or
disprove: f (A) is an open subset of Rm .
(b) Let B

Problem Set 8
MATH-23a/MATH-E23a, Fall 2012
due 10/31/2012, in class
This PSet is more mechanical than others we have had - I hope you all still remember how to do
computations after spending lots of time in the world of abstract mathematics!
(1) (borrowe

Problem Set 9
MATH-23a/MATH-E23a, Fall 2012
due 11/7/2012, in class
(1) Do the following exercises in Hubbard: 2.3.6, 2.3.8
(2) Do the following exercises in Hubbard: 2.4.2, 2.4.9, 2.4.12
(3) Do the following exercises in Hubbard: 2.5.2, 2.5.6
(4) Do the

MATHEMATICS 23a/E-23a, Fall 2015
Linear Algebra and Real Analysis I
Module #1, Week 4 (Eigenvectors and Eigenvalues)
Author: Paul Bamberg
R scripts by Paul Bamberg
Last modied: October 1, 2015 by Paul Bamberg (simplied page 25)
Reading
Hubbard, Section 2