The RangeNullspace Decomposition of Cn
Math 422
Denition 1 Let A be an n n matrix. The range (or column space) of A; denoted by R (A) ; is the subspace
of Cn spanned by the columns of A: The nullspace of A, denoted by N (A) ; is the solution space of the
GLOSSARY: A DICTIONARY FOR LINEAR ALGEBRA
Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node
i to node j; otherwise aij = 0. A = AT for an undirected graph.
Affine transformation T (v ) = Av + v 0 = linear transformati
Starting with Two Matrices
Gilbert Strang, Massachusetts Institute of Technology
Imagine that you have never seen matrices. On the principle that examples are amazingly powerful, we
study two matrices A and C . The reader is requested to be exceptionally
18.06 Spring 2009 Exam 2 Practice
General comments
Exam 2 covers the rst 18 lectures of 18.06. It does not cover determinants (lectures 19 and 20). There will also be no
questions on graphs and networks. The topics covered are (very briey summarized):
1.
Appendix G: Sample Laboratory Report
There is no set length for a problem report but experience shows that good reports are typically three
pages long. Graphs and photocopies of your lab journal make up additional pages. Complete reports
will include the
18.06
Professor Johnson
Quiz 1
October 3, 2007
SOLUTIONS
1 (20 pts.)
Find all solutions to the linear system
x + 2y + z 2w = 5
2x + 4y + z + w = 9
3x + 6y + 2z w = 14
Solution:
We perform
1
2
3
elimination on the augmented matrix:
2 1 2 5
1 2 1 2 5
1 2 1
18.06
QUIZ 1
March 05, 2007
Your PRINTED name is: SOLUTIONS
Please circle your recitation:
(1)
M2
2131 A. Osorno
(2)
M3
2131 A. Osorno
(3)
M3
2132 A. Pissarra Pires
(4) T 11 2132 K. Meszaros
(5) T 12 2132 K. Meszaros
(6)
T1
2132 Jerin Gu
(7)
T2
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18.06
Professor Strang
Quiz 1
February 28, 2005
Grading
1
Your PRINTED name is:
SOLUTIONS
2
3
4
1 (26 pts.)
Suppose A is reduced by the usual row
14
R= 0 0
00
operations to
02
1 2 .
00
Find the complete solution (if a solution exists) to this system invol
TABLE OF CONTENTS
Introduction
Laboratory I: Electric Fields and Forces
Simulation Problem #1: Electric Field Vectors
Problem #2: Electric Field from a Dipole
Problem #3: Gravitational Force on the Electron
Problem #4: Deflection of an Electron Beam by an