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Course: MATH 2431, Fall 2008
School: U. Houston
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Word Count: 660

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2431 Math Here are some of the topics discussed in this course: vectors in Rn , matrices addition, multiplication by a scalar product of matrices, and matrix vector; properties (linear maps = matrix maps) norm and dot product of vectors, angles and areas systems of linear equations: Ax = b row echelon form via Gauss elimination reduced row echelon form is unique nding solutions (a basis of solutions if b...

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2431 Math Here are some of the topics discussed in this course: vectors in Rn , matrices addition, multiplication by a scalar product of matrices, and matrix vector; properties (linear maps = matrix maps) norm and dot product of vectors, angles and areas systems of linear equations: Ax = b row echelon form via Gauss elimination reduced row echelon form is unique nding solutions (a basis of solutions if b = 0) rank consistent and inconsistent systems if consistent: # of params = # unknowns rank A if b = 0, dimension of solution space = # unknowns rank A superposition (3.4) solutions for low-dimensional systems matrices, linearity, inverses linear maps ( = matrix maps) linear maps in R2 superposition (3.4, 4.7) computing inverses (via row reduction) solving Ax = b with A1 2 x 2 determinants and inverses; area(A(P )) = | det(A)| area(P ) systems of ODEs one ODE, initial value problems (x = f (x, t), x(t0 ) = x0 ) autonomous ODEs (x = g(x)); equilibria (g(x) = 0) and stability of hyperbolic equilibria (g (x0 ) = 0) x = x = x(t) = x0 et 2 x 2 systems (see much more in Chapter 6); sinks, sources, saddles graphic representations: time-series and phase-space portraits vector spaces denition subspaces (V vector space, W V ; W subspace W closed under addition and multiplication by scalars) e.g.: null(A) = {x Rn | Ax = 0}, {X (C 1 )n | X = CX} span of a family of vectors spanning sets linear independence dimension, bases Theorem 5.5.3, Corollary 5.6.7 1 dimension of a span and of a null-space: dim(span{w1 , . . . , wk }) = rank(M t ) = rank(M ), M = (w1 | . . . |wk ) dim(null(A)) + rank(A) = # of columns (i.e., # of variables) planar ODEs: X = CX, C an n n matrix ASIDE. general ODEs: X = F (X); F has continuous derivatives = uniqueness solutions of to IVP dimension of space of solutions = n (size of C), hence need n linearly independent solutions and use superposition solutions {X1 (t), . . . , Xn (t)} are linearly independent if so is the family of vectors {X1 (0), . . . , Xn (0)} direct method and matrix exponentiation direct method (6.2): three cases 1 , 2 real, two linearly independent eigenvectors 1 , 2 complex conjugate 1 = 2 real, one linearly independent eigenvector (use generalized eigenvector) matrix exponentiation: nd similarity to normal form (Thm. 6.5.5) determinants and eigenvalues (for n n matrices) properties: det(A) for A triangular, det(A) = det(At ), det(AB) = det(A) det(B) computing determinants: reduction to row echelon form or cofactor expansion A invertible det(A) = 0 characteristic polynomial pA () = det(A I) eigenvalues: roots of pA () = 0 eigenvectors: nonzero vectors with Av = v (hence is an eigenvalue) det(A) = 1 2 . . . n , trace(A) = 1 + + n det(A) = 0 no eigenvalue equal to zero A invertible = eigenvalues of A1 are the inverses of the ei...

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U. Houston - MATH - 2431
Here is a selection of problems from the sections covered by the second exam. The even-numbered problems that were already assigned as homeworks are denoted by a *. 4.3: 7, 9, 12*, 13, 15 4.5: 1, 3, 4*, 5 4.6: 4, 5 4.7: 7 4.8: 7 4.9: 1, 3 5.1: 3, 5,
U. Houston - MATH - 2431
Here is a selection of problems from the sections covered by the second exam. Some of the problems were already assigned as homework. Note that some of the problems can be solved with more than one method (as we progressed in the book, we learned mor
U. Houston - MATH - 2431
Math 2431 Here are some of the topics discussed so far: vectors in Rn , matrices addition, multiplication by a scalar product of matrices, and matrix vector; properties (linear maps = matrix maps) norm and dot product of vectors, angles and area
U. Houston - MATH - 2431
Math 2431 Here are some of the topics discussed before the rst exam: vectors in Rn , matrices addition, multiplication by a scalar product of matrices, and matrix vector; properties (linear maps = matrix maps) norm and dot product of vectors, an
U. Houston - MATH - 6298
How to write slides(an incomplete guide) Nov. 30, 2000These are the notes for the title page Explain how to orient the slides on the projector: the speaker should be able to read them, as he faces the public. It is not recommended for the spea
U. Houston - MATH - 6298
Name: Student #:MATH 7890 Fall 2007Exam II Analytic geometry Oct. 13, 20071. Find the intersection of the lines x+y =4 and x-y =32. Write the equation of the following curves: (a) The circle with radius 5 and center (1, 7).(b) A circle tange
U. Houston - MATH - 6298
A L TEX 2 for authorsA c Copyright 19951999, L TEX3 Project Team. All rights reserved.24 September 1999Contents1 Introduction A A 1.1 L TEX 2 -The new L TEX release . . . . A A 1.2 L TEX3-The long-term future of L TEX 1.3 Overview . . . . . . .
U. Houston - MATH - 7350
Geometry of Manifolds a previewThese notes are an informal presentation of a few Differential Geometry topics.Why manifolds?The &quot;simplest&quot; higher dimensional objects are the vector spaces. manifolds versus vector spaces curves versus straight
U. Houston - MATH - 1431
Gxy xaY 9 x# ~ w C m x x 9 ux# w C B Y x y % x 9 ux# w 6 6 6' # 6 4 d h h t`1Dr h t`\$a3VvaSRP } g ig t1 h2 32t1SRR g 3Pc1T&quot;l\$a3VQt`DSj3P(R h` tP d 1g h t`Pi762 (j54&quot;S21` 32 r i1 ' Y\$u j j 2 1 ` 1 ` 4 h u P P bR ` 1 1 1 ` P0 # ! ) 46515`SR d
U. Houston - MATH - 7350
Homework 1Updated January 26, 2009; footnotes added Feb. 3, 2009Due date: February 5, 2009 (Thursday) DO NOT HAND IN: Please think about these problems, but do not hand in the solutions. Exercises 1.1, 1.2 and 1.3. TO HAND IN: Please write soluti
U. Houston - COUPLED - 60
Driving directionshttp:/www.math.uh.edu/~torok/coupled60/incoming_directions-driving.Driving directions to UH Hilton, Best WesternUse these directions for: driving to your hotel (UH Hilton or Best Western) on arrival driving between UH Hilton (s
U. Houston - MATH - 3339
University of Houston Mathematics Department Math 3339 StatisticsPrerequisite: Math 3338 Course Description: Sampling; estimation and hypothesis testing; regression; analysis of variance; exploratory techniques. Text: Probability and Statistical Inf
U. Houston - COUPLED - 60
SHUTTLE DIRECTIONS FROM AIRPORTShttp:/www.math.uh.edu/~torok/coupled60/incoming_directions-shuttle_.SHUTTLE DIRECTIONS FROM AIRPORTSHOTELS: Best Western Hotel 2929 Southwest Freeway (I-59) Houston, TX 77098 phone: 713-528-6161 Hilton University
U. Houston - MATH - 2433
CHAPTER 14 SECTION 14.1 3. 4. 7. 9. dom (f ) = the set of all points (x, y) except those on the line y = -x; dom (f ) = the set of all points (x, y) other than the origin; range (f ) = (-, 0) (0, )range (f ) = (0, )dom (f ) = the first and third
U. Houston - MATH - 2433
SUMMARY: CHAPTERS 16 and 17CHAPTER 16 MULTIPLE INTEGRALS I. DOUBLE INTEGRALS a. Denition: Let f = f (x, y) be continuous on the rectangle R : a x b, c y d. Let P be a partition of R and let mij and Mij be the minimum and maximum values of f on
U. Houston - MATH - 2433
N(t) =T (t) T (t)=|x y x y | [(x )2 + (y )2 ]3/2 dT dT/dt = ds ds/dt ds d2 s T+ dt2 dt va (ds/dt)32=andds/dt = v is the speeda=N=fu (x) =f (x) uz z0 =g g (x0 , y0 )(x x0 ) + (x0 , y0 )(y y0 ) x y g (x0 , y0 )t, x g (x0
U. Houston - MATH - 2433
Chapter 12:VECTORS1. Geometry: Let P1 (x1 , y1 , z1 ) and P2 (x2 , y2 , z2 ) be points in 3-space: A. Distance Formula: d(P1 , P2 ) = (x2 x1 )2 + (y2 y1 )2 + (z2 z1 )2 .B. Midpoint Formula: The midpoint of the line segment joining P1 and P2
U. Houston - MATH - 2433
CHAPTER 15 SECTION 15.11. 5. 6. 9. 11. 12.f = (6x - y) i + (1 - x) j3.f = exy [ (xy + 1) i + x2 j]f = 2y 2 sin(x2 + 1) + 4x2 y 2 cos(x2 + 1) i + 4xy sin(x2 + 1) j f= 2y 2x i+ 2 j 2 +y x + y2x2f = (z 2 + 2xy) i + (x2 + 2yz) j + (y 2 + 2z
U. Houston - MATH - 6298
Here are a few more often used commands, from a list I made for myself.Lines starting with - denote commands that do not have a key-sequenceshorcut.REMARKS: &quot;ESC x&quot; is equivalent to &quot;M-x&quot;. All commands can also be invoked by &quot;M-x (command-n
U. Houston - MATH - 6298
\documentclass[10pt]{article}\usepackage{amssymb, amsmath}\title{A Short Paper}\author{Al G. Ebra}\date{\today}\newtheorem{thm}{Theorem}[section]\newtheorem{lemma}[thm]{Lemma}\newtheorem{prop}[thm]{Proposition}\newtheorem{defn}[thm]{Definit
U. Houston - MATH - 6298
function z=sc(x,y)% sc(x,y)=sin(x)*cos(x+y)z=sin(x)*cos(x+y);
U. Houston - MATH - 6298
% DRAW SECTIONS (same as CONTOUR, but the curves are lifted to their z-value)% NOTE: The computation of z in this case could be% achieved faster with meshgrip and Matlab's array% operations, which are usually quicker than the cycles:% [x1,
U. Houston - MATH - 6298
function y=iteration(funct,point,tol)% y=iteration(FUNCT,POINT,TOL) applies the function FUNCT to POINT%and iterates until the value does not change by more%than TOL;%% FUNCT should be a string or a function handle%% default value of TOL is
U. Houston - MATH - 6298
function y = slow(x)% slow(x)=(a+x)/(1+x), a slow iteration to find the square root of a%% The function to be iterated is%% y= (a+x)/(1+x)%% The value a has to be declared as 'global' and initialized in workspace (the &quot;main&quot; program)globa
U. Houston - MATH - 6298
% DRAW A CURVE IN 2 DIMENSIONS, ETC.% choose the range (pi=3.14.) and points where to evaluate:t=-pi:.01:pi;% plot (this will open a &quot;figure&quot;, unless one was in use already)plot(sin(3*t),2*cos(5*t)% NOTE that Matlab offers ways to adju
U. Houston - MATH - 6298
Name: Student #:MATH 7890 Fall 2007Exam II Analytic geometry Oct. 13, 20071. Find the intersection of the lines x+y =4 and xy =32.Write the equation of the following curves: 1. The circle with radius 5 and center (1, 7).2. A circle tangent
U. Houston - MATH - 6298
GNU Emacs Reference Card(for version 20)Starting EmacsTo enter GNU Emacs 20, just type its name: emacs To read in a file to edit, see Files, below.Leaving Emacssuspend Emacs (or iconify it under X) exit Emacs permanently C-z C-x C-cFilesrea
U. Houston - MATH - 6298
An exampleAnonymous September 21, 2008A A Before we start, here are the correct type-settings of TEX, L TEX and L TEX 2 (notice the empty string {} or extra space \ we had to use; otherwise, we obtain A TEXand L TEX). It is better to use {}, since
U. Houston - CH - 12
'Kyphosis','Nokypho','age','kyph1'12,1,12,115,1,15,142,2,42,152,8,52,159,11,59,173,18,73,182,22,82,191,31,91,196,37,96,1105,61,105,1114,72,114,1120,81,120,1121,97,121,1128,112,128,1130,118,130,1139,127,139,1139,131,139,1157,140,157,
U. Houston - CH - 11
'Vibrat','Source','Material'13.1,1,'S'13.2,1,'S'15,1,'A'14.8,1,'A'14,1,'P'14.3,1,'P'16.3,2,'S'15.8,2,'S'15.7,2,'A'16.4,2,'A'17.2,2,'P'16.7,2,'P'13.7,3,'S'14.3,3,'S'13.9,3,'A'14.3,3,'A'12.4,3,'P'12.3,3,'P'15.7,4,'S'15.8,4,'S'13.7,
U. Houston - CH - 12
'Success','Failure','Exper','success1'8,4,8,113,5,13,114,6,14,118,6,18,120,7,20,121,9,21,121,10,21,122,11,22,125,11,25,126,13,26,128,15,28,129,18,29,130,19,30,132,20,32,1,23,4,0,27,5,0,6,0,6,0,7,0,9,0,10,0,11,0,11,0,13,0,15,0
U. Houston - CH - 11
'Brand1','Brand2','Brand3','Brand4','folacin','brand'7.900000095367432,5.699999809265137,6.800000190734863,6.400000095367432,7.900000095367432,16.199999809265137,7.5,7.5,7.099999904632568,6.199999809265137,16.599999904632568,9.800000190734863,5,7.
U. Houston - CH - 12
'x1','x2','y','x1x2'.6,200,90.6,120.6,250,82.7,150.6,400,58.7,240.6,500,43.2,300.6,600,25,3601,200,127.1,2001,250,112.3,2501,400,19.6,4001,500,17.8,5001,600,9.1,6002.6,200,53.1,5202.6,250,52,6502.6,400,43.4,10402.6,500,42.4,13002.6,600
U. Houston - CH - 11
'Fe','form'20.5,128.100000381469727,127.799999237060547,127,128,125.200000762939453,125.299999237060547,127.100000381469727,120.5,131.299999237060547,126.299999237060547,224,226.200000762939453,220.200000762939453,223.700000762939453,2
U. Houston - CH - 11
'force','connect','angle'45.29999923706055,1,'0 deg'44.099998474121094,1,'2 deg'42.70000076293945,1,'4 deg'43.5,1,'6 deg'42.20000076293945,2,'0 deg'44.099998474121094,2,'2 deg'42.70000076293945,2,'4 deg'45.79999923706055,2,'6 deg'39.59999847
U. Houston - CH - 07
'VERBIQ','mf'117,'male'103,'male'121,'male'112,'male'120,'male'132,'male'113,'male'117,'male'132,'male'149,'male'125,'male'131,'male'136,'male'107,'male'108,'male'113,'male'136,'male'114,'male'114,'female'102,'female'113,'female'
U. Houston - CH - 12
'height','foot','wingspan','leverage'63,9,62,.2398663,9,62,.2398665,9,64,.22823664,9.5,64.5,.22362568,9.5,67,.19641869,10,69,.08367671,10,70,.26218268,10,72,.06720768,10.5,70,.18708872,10.5,72,.15195973,11,73,.14327973.5,11,75,.16871970,
U. Houston - CH - 14
'Stiff','Length'309.20001220703125,'4&quot;'309.70001220703125,'4&quot;'311,'4&quot;'316.79998779296875,'4&quot;'326.5,'4&quot;'349.79998779296875,'4&quot;'409.5,'4&quot;'331,'6&quot;'347.20001220703125,'6&quot;'348.8999938964844,'6&quot;'361,'6&quot;'381.70001220703125,'6&quot;'402.1000061035156
U. Houston - CH - 11
'stiff','length'309.20001220703125,4409.5,4311,4326.5,4316.79998779296875,4349.79998779296875,4309.70001220703125,4402.1000061035156,6347.20001220703125,6361,6404.5,6331,6348.8999938964844,6381.70001220703125,6392.3999938964844,8366.2
U. Houston - CH - 09
'TIPpc'14.2120.2420.114.9415.6915.0412.0420.1617.8516.3519.1220.3715.2918.3927.5516.0110.9413.5217.4214.4829.8717.9219.7422.7314.5615.1616.0916.4219.0713.7413.4616.7919.0319.1919.2312.3916.8918.9313.5617.711.4
U. Houston - CH - 10
'strength','treat','fusion'2748,1,'no fusion'2700,1,'no fusion'2655,1,'no fusion'2822,1,'nofusion'2511,1,'nofusion'3149,1,'nofusion'3257,1,'nofusion'3213,1,'nofusion'3220,1,'nofusion'2753,1,'nofusion'3027,2,'fused'3356,2,'fused'3359,2,'f
U. Houston - CH - 12
'time','glucose'2,42,3.59999990463256845,3.7000000476837167,412,3.79999995231628413,417,5.09999990463256818,3.900000095367431623,4.40000009536743224,4.30000019073486326,4.30000019073486328,4.40000009536743229,5.80000019073486330,4.30000
U. Houston - CH - 11
'Yield','Speed','Formulat'189.6999969482422,60,1188.60000610351562,60,1190.10000610351562,60,1185.10000610351562,70,1179.39999389648438,70,1177.3000030517578,70,1189,80,1193,80,1191.10000610351562,80,1165.10000610351562,60,2165.89999389648
U. Houston - CH - 08
'length'1941601762031871631621831521771771511731881791941491651861871871771871861871731361501731731361531521491521801861661741761981932181731441481741631841551511722161492072122161661901651
U. Houston - MATH - 2303
Math 2303 Concepts in AlgebraSection 19410 Room 201 Garrison TTh 10 a.m. 11:30 a.m.Instructor:Marjorie Marks Email address: mmarksc@math.uh.edu Website: www.math.uh.edu/~mmarksc Conference hours: MW 12:15 p.m. 2:15 p.m. in 222 Garrison, or by
U. Houston - MATH - 1314
Math 1314 Lesson 18 Area and the Definite Integral We are now ready to tackle the second basic question of calculus the area question. We can easily compute the area under the graph of a function so long as the shape of the region conforms to someth
U. Houston - MATH - 1314
Math 1314 Lesson 16 Antiderivatives So far in this course, we have been interested in finding derivatives and in the applications of derivatives. In this chapter, we will look at the reverse process. Here we will be given the answer and well have to
U. Houston - MATH - 1330
Review for Test 1 1. Find intercepts: 5x - 2y = 20 y = 3x - 1 y = x 2 + 7x + 12 y = - x 2 + 4x - 22.Find intercepts:3.Find intercepts:4.Complete the square: identify vertex sketch graph5.List shifting instructions: y=- x+2y = 3- x
U. Houston - MATH - 1314
HelloYou re receiving this email because you are currently enrolled (as of Dec. 29, 2008) in Math 1314, section 19290, the online section of Math 1314, for spring semester 2009. If you do not wish to be enrolled in an online section for this course,
U. Houston - MATH - 2303
U. Houston - MATH - 1330
Math 1330 Review for Final Exam You should be able to do all of these for any function we give you: Find domain Find range Find x and y intercepts Find any vertical, horizontal or slant asymptotes State where a graph crosses its horizontal asy
U. Houston - MATH - 1330
U. Houston - MATH - 1330
Potential Poppers 6.3 Note that if the problem is from the exercises, the problem number is in parenthesis. Is3 a solution to sin 3 x - 4sin 2 x - 2 sin x = 3 ? 2tan x = -1 csc x = -2 2 sin 2 x - 5sin x - 3 = 0 cos2 x = 2 cos x - 1(3) (7
U. Houston - MATH - 1314
Math 1314 Lesson 18 Area and the Definite Integral We are now ready to tackle the second basic question of calculus the area question. We can easily compute the area under the graph of a function so long as the shape of the region conforms to someth
U. Houston - MATH - 2303
Math 2303 January 22 From last class: Example 1: Write 54,221 as an Egyptian number.Example 2: Write the Egyptian number as a Hindu-Arabic number.Roman Numerals: An additive system with a couple of twists. Here are the numerals:Roman I V X L C
U. Houston - M - 1313
Math 1313 Section 19280Popper 03 Form AUse the following information for all questions: The Mathemagic Toys toy store produces widgets. The production cost of one widget is \$2.15, and it sells for \$5.45. The company has fixed costs of \$660,000. 1
U. Houston - M - 1300
Math 1300Suggested HomeworkSpring 2009The following homework is a suggestion only. It is not assigned and will not be graded if turned in. Do the suggested problems if you are having difficulties in any particular section. SECTION 1.1 1.2 1.3 1
U. Houston - M - 1300
Math 1300Assigned HomeworkSpring 2009Use only one red scantron per homework assignment. Homework must be turned in on red scantron pages. If two homework assignments are due on the same day, you must use two separate red scantrons (no staples).
U. Houston - M - 1300
Math 1300 Greatest Common Factor and Factoring by GroupingSection 4.1 Notes(Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplies by another number, variable, monomial, or polynomial to obtain a
U. Houston - M - 1314
M 1314lesson 2 Math 1314 Lesson 2 One-Sided Limits and Continuity One-Sided Limits1Sometimes we are only interested in the behavior of a function when we look from one side and not from the other. Example 1: Consider the function f ( x) =x x .
U. Houston - M - 1300
Math 1300Section 1.7 NotesSolving Linear Inequalities An inequality is similar to an equation except instead of an equal sign = you find one of the following signs: &lt;, , &gt;, or . Now &gt; and &lt; are strict inequalities, and and are inequalities that