MA 1B RECITATION 03/05/15
1. Linear Algebra from 10,000 feet
This is the last recitation for the quarter, so I think its worthwhile to look back
at the course and look at what the main ideas are and what are the details. My own
personal perspective is tha
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 8
Due: 10am, Monday, March 10, 2014
1. Problem 14 on page 126 of the text.
2. Prove that for each 2 by 2 matrix A over C, exactly one of the following holds:
(1
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b; Solutions to Homework Set 5
Due: February 18, 2014
1. Observe rst that for a, b R and m a positive integer,
m1
am1i bi .
am bm = (a b)m (a, b), where m (a, b) =
(!)
i=0
2
Further 1 (a,
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b; Solutions to Homework Set 4
Due: February 3, 2014
1. For T L(V ) let m(T ) = mX (T ) be the matrix of T with respect to X . We
rst calculate m(f ) and m(f 2 ). Now f (e1 ) = f (1, 0) =
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b; Solutions to Homework Set 3
Due: January 27, 2014
1. (a) (ST )(x, y, z ) = S (T (x, y, z ) = S (x, x + y, x + y + z ) = (x + y + z, x + y, x).
Similarly (T S )(x, y, z ) = T (z, y, x) =
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b; Solutions to Homework Set 2
Due: January 21, 2014
1. We will see later in the course that the solution space of m independent homogenous
linear equations in n m variables is of dimensio
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b; Solutions to Homework Set 1
Due: January 13, 2014
1. We must show W is nonempty and closed. By Remark 2 in class, 0 U for each
U U , so 0 W . Thus W = . Let x, y W and a, b F . Then x,
Math1b-Practical, Winter 2008-2009
Homework 7-solutions
1. solution. For some , if I A is singular, then is an eigenvalue of A. So we can
try 17 and 5. It is obvious that 17 and 5 are eigenvalues of A. For 17, we can nd all
linearly independent eigenvecto
mma5.txt
Mathematica Session 5 - Determinants
Math 1b Practical
February 9, 2009
(The numbers are out of order on the attached MMA session.)
We show the determinants of general 2x2, 3x3, 4x4, and 5x5
matrices in Out[12], Out[14], Out[2], and Out[4], respe
Math1b-Practical, Winter 2008-2009
Homework 8-solutions
(Notation: the notation a := b means that we are dening a by setting it equal to b.)
1. (i) We must nd an orthonormal basis u1 , u2 , . . . , u6 for R6 consisting of eigenvectors of S . We know from
Math 1b Problem Set 7
Revised Feb. 26, 2009
Due 3:00pm, Monday, March 2, 2009
Do Problem 1 by hand. You should use Matlab, Mathematica, Maple, or a calculator
for the next two problems. Report the answers to two or three signiant digits; thats good
enough
Math 1b Basic solutions; nonnegative solutions
January 12, 2009
1. Basic solutions to a system
We use matrices to model systems of linear equations.
7 2
7
1 4 3
3
1 4
(1)
To solve the system, we want to nd an equivalent system of equations so that the co
Math 1b Problem Set 5 Solutions
Winter 2009
Solution 1
(i) By Gram-Schmidt orthogonalization process, take v1 = v1 = (1 1 1 1 1).
v 2 , v1
Then v2 = v2
v = v2 3v1 = (2 1 0 1 2).
v 1 , v1 1
By solving the above equations, we get v1 = v1 and v2 = 3v1 + v2
Math 1b Problem Set 7
Due 3:00pm, Monday, March 2, 2009
Do Problem 1 by hand. You should use Matlab, Mathematica, Maple, or a calculator
for the next two problems. Report the answers to two or three signiant digits; thats good
enough.
1. What are the eige
Math 1b Problem Set 6
Due 3:00pm, Monday, February 23, 2009
[Since we missed a day, this is a shorter problem set, worth 70 points.]
1. (25 points) (i) Use Cramers Rule to solve Ax = b for x where
20
1
A = 0 3 2
3 1 1
for the following four values of b (
Math 1b Problem Set 4
Due 3:00pm, Monday, February 2, 2009
1. Let U be the span of the three vectors
(2, 1, 1, 3),
(1, 0, 2, 1),
(5, 1, 5, 0).
(i) Find a basis for U . What is the dimension of U ? Show some work.
(ii) Find a basis for U . What is the dime
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 1
Due: 10am, Monday, January 13, 2014
Read the rst 9 sections of Chapter 1 of Apostol. That is read pages 3 through 13.
You can collaborate on the problems as l
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 2
Due: Noon, Tuesday January 21, 2014
Read sections 1 through 5 and 9 through 16 of Chapter 2. Skip sections 6 and 7
of Chapter 2 as the discussion of inverses
MA 1B RECITATION 02/26/15
1. Inner Product Spaces, Norms, Orthogonal/Orthonormal Vectors
Recall that a real vector space V is said to have an inner product (, ) : V V R
if the function (, ) taking ordered pairs to the scalar eld satises
(x, y) = (y, x)
MA 1B RECITATION 02/19/15
1. Eigenvectors and Eigenvalues of a Matrix
Recall that given a matrix A, an eigenvalue of A is a constant such that
Av = v
for some nonzero vector v, called a -eigenvector of A.
To me, the eigenvalues represent the essence of a
MA 1B RECITATION 02/12/15
1. Determinants
1.1. What do determinants represent? When introduced at this stage of the
course, determinants seem like mysterious objectsjust a formula that you apply
blindly to matrices to get some number out of them. However,
MA 1B RECITATION 02/05/13
1. Announcements
The midterm is now available. If you dont have a copy for some reason, pick
one up from the Math Oce on the second oor Sloan. The midterm is due Sunday
night at the same time as the homework.
The midterm review w
MA 1B RECITATION 01/29/15
1. Announcements
The midterm will be handed out next week.
Midterm review will be in Sloan 151 either next Thursday or Friday at 8pm in
Sloan 151. Please answer the poll to indicate which of these times work for you.
Next week wi
MA 1B RECITATION 01/22/15
1. Feedback
We have been using the following terminology for linear transformations and
such without explanation, but it seems like not everybody is familiar with it, so
lets quickly recall the denitions.
We say that a map of set
MA 1B RECITATION 01/14/15
1. Warm-up Question
Consider the vector space of 3-by-3 matrices and consider the matrices in it that
are magic squares, where the rows, columns, and diagonals all add up to the same
number, for example
6 1 8
7 5 3 .
2 9 4
We can
MA 1B RECITATION 01/08/15
1. Introduction
Hi, Im Brian, and Ill be your TA this quarter. Heres some basic info
My email: bhwang@caltech
My oce: 158 Sloan
My oce hours: 4pm Sunday afternoon, or by appointment.
Section website: http:/hwang.caltech.edu/m
MTH 203B: Geometry B |Problem Set 3
Name: _ Date: _
Directions: Complete the problems below. Showing your work is more important than just a correct answer
without work. Answers need to clearly shown and units given when appropriate. SHOW ALL WORK.
ALL pr
Chapter 3. Determinants.
Let F = R or C and Mn the set of all n by n matrices over F . The text denes
the determinant as a function from Mn into F satisfying certain axioms. It goes on to
derive various properties of such a function and to show that there
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 8
Due: 10am, Monday, March 10, 2014
1. Problem 14 on page 126 of the text.
2. Prove that for each 2 by 2 matrix A over C, exactly one of the following holds:
(1