LECTURE 1.
Start with Strangs elegant description of errors and blunders. Errors are
unavoidable aspects of any computation, whether because of a computers
inherent limitations or our own human capacity for mistakes. Blunders are
avoidable mistakes that c
LECTURE 1
LECTURE 2
0. Distinct eigenvalues
I havent gotten around to stating the following important theorem:
Theorem: A matrix with n distinct eigenvalues is diagonalizable.
Proof (Sketch) Suppose n = 2, and let 1 and 2 be the eigenvalues,
v1 , v2 the e
LECTURE 1
I. Column space, nullspace, solutions to the Basic Problem
Let A be a m n matrix, and y a vector in Rm . Recall the fact:
Theorem: Ax = y has a solution if and only if y is in the column space
R(A) of A.
Now lets prove a new theorem which talks
LECTURE 1 I. Inverse matrices
We return now to the problem of solving linear equations. Recall that
we are trying to nd x such that
Ax = y.
Recall: there is a matrix I such that
Ix = x
for all x Rn . It follows that
IA = A
for all n n matrices A.
For the
Linear Algebra Notes
Brad Lackey
D EPT OF M ATHEMATICS , U NIV OF H ULL , H ULL HU6 7RX, UK
E-mail address: B.Lackey@maths.hull.ac.uk
Contents
Preliminaries
1. Arithmetic
2. Algebra
3. Geometry
4. Calculus
5. Logic and Set Theory
1
1
3
3
4
4
Part I. Appli
MATH. 513. JORDAN FORM
Let A1 , . . . , Ak be square matrices of size n1 , . . . , nk , respectively with entries in a eld F . We
dene the matrix A1 . . . Ak of size n = n1 + . . . + nk as the block matrix
A1
0
0 .
0
0
A2
0 .
0
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Math. 513. Linear Algebra, Winter 2009/10
Instructor: Igor Dolgachev, 3064 EH, 763-0151 (oce)
e-mail: idolga@umich.edu
Textbook S. Friedberg, A. Insel, L. Spence Linear Algebra, 4d edition, Prentice Hall, 2003.
+handouts
Oce Hours: M 10-11, W 11-12, F 3-4
Math. 513. Fall 2003. Midterm Exam
Part I. True False (20 pts) Circle the number which represents the true statement (no expanation needs
to be given, no partial credit is given).
1. The inverse of a bijective linear map of linear spaces is a linear map.
Math. 513. Homework 1
In the following Q, R, C denote the led of rational, real and complex numbers. Also Fp denotes the
Galois eld of p elements.
1. Check whether the following set F forms a eld with respect to given operations.
(a) F = cfw_a + b 2 R : a
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