# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

8 Pages

### s08hw5

Course: MATH 205, Spring 2006
School: Lehigh
Rating:

Word Count: 585

#### Document Preview

5: Week 2.2 Subspaces/Spanning (continued) 2.3 Independence/Bases 2 Linear independence/Bases and Dimension Vectors v1 , v2 , . . . , vk in a vector space V are linearly dependent if there are scalars c1 , c2 , . . . , ck so 0 = c1 v1 + c2 v2 + + ck vk , with at least one cj = 0. Such a vector equation is said to be a relation of linear dependence among the v1 , v2 , . . . , vk . A collection of vectors for...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Pennsylvania >> Lehigh >> MATH 205

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
5: Week 2.2 Subspaces/Spanning (continued) 2.3 Independence/Bases 2 Linear independence/Bases and Dimension Vectors v1 , v2 , . . . , vk in a vector space V are linearly dependent if there are scalars c1 , c2 , . . . , ck so 0 = c1 v1 + c2 v2 + + ck vk , with at least one cj = 0. Such a vector equation is said to be a relation of linear dependence among the v1 , v2 , . . . , vk . A collection of vectors for which the only solution of the vector equation is the trivial solution, c1 = 0, . . . , ck = 0 is said to be linearly independent. Examples: (1) i, j in R2 ; and (2) {i, j, k} in R3 ; are linearly independent. Verification: c1 i + c2 j + c3 k = c1 (1, 0, 0) + c2 (0, 1, 0) + c3 (0, 0, 1) = (c1 , c2 , c3 ) = (0, 0, 0) = 0, only when c1 = 0, c2 = 0, c3 = 0. Our main objective is the definition of basis. Vectors v1 , v2 , . . . , vk in a vector space V that are are both (1) a spanning set for V and (2) linearly independent are called a basis for V. So (1) i, j are a basis of R2 ; and (2) {i, j, k} is a basis of R3 . A vector space has many bases, but the main fact is that if v1 , v2 , . . . , vk is a basis of V and w1 , w2 , . . . , wj is a second basis of V, then the number of vectors is the same: k = j. The number of vectors in a basis of V is called the dimension of V . We start with verifications of linear independence in subspaces of Rn 4 Problem Determine whether the vectors (1, 2, 3), (1, -1, 2) and (1, -4, 1) are LI or LD in R3 . If they are LD find a relation of linear dependence. Solution [again; same method!] Write the vector equation as an equation in column vectors, then 1 1 1 0 0 = c1 2 + c2 -1 + c3 -4 1 2 3 0 1 1 1 c1 = 2 -1 -4 c2 . 3 2 1 c3 Observe that this is the homog linear system with coef matrix A = (v1 v2 v3 ). For n-by-n systems the system has a unique solution exactly when the determinant is non-zero. 1 We have 2 3 1 -1 2 1 1 -4 = 0 1 0 1 -3 -1 1 -6 = 0 -2 so the system has nontrivial solutions and the vectors are LD. To find a relation we continue the reduction 1 0 -1 2 , A 0 1 0 0 0 so a spanning vector is (1, -2, 1), and 0 = v1 - 2v2 + v3 is a nontrivial relation of LD. 6 Example: Show that the vectors v1 = (1, 1) v2 = (1, -1) are a basis of R2 . Solution. We know that i, j are a basis of R2 ; so that the dimension of R2 is . . . . We check that v1 = (1, 1) and v2 = (1, -1) are linearly independent (LI) since 1 1 1 -1 = -2 = 0. Checking that v1 and v2 span R2 by expressing each v = (x, y) R2 as a linear combination (as we did for i, j) takes a little bit of effort, and involves fractions. Checking that each vector equation v = c1 v1 + c2 v2 has a solution (without finding c1 , c2 ) is easier. (why?) But the main fact has a practical version that saves any computation, or even very much further thought. If v1 and v2 did not span R2 there would be a 3rd vector v3 not in the span of v1 , v2 . But that would make v1 , v2 , v3 linearly independent. We could consider adding further vectors, each new one giving a 8 larger collection of indep vectors; but we already have too many, as we've shown dim(R2 ) 3. So v1 , v2 have to span (for free, given the main fact), so v1 , v2 is a basis for R2 . This is the use of the text's Theorem 2.12.(1) If the dim(V ) = n for a vector space V, then any LI set of n vectors is a basis of V.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Lehigh - MATH - 205
Week 6, 7: 2.5 Wronskian 3.1 Intro, Slope Fields, verify solution 3.2 Separable DE 3.3 Exact Equations 3.4 Linear Equations 3.6 Cooling/Mixing2Problem 1: Verify that the function y = c1 x is a y solution of y = 2x Solution: Compute y and check.1
Lehigh - MATH - 205
Math 205, Spring 2008 B. Dodson Text - Peterson-Sochacki Office hours: M,W: 10:20-11:20, Tu: 11:30 - 12:30 and by appt. email: bad0@lehigh.edu -1. Course Info 2. Homework 1: 1.1: 1-6, 17, 18, 21 Due Friday -An m n matrix A is an array with m horizon
Lehigh - CSC - 340
CSc 340 Test 1 Wednesday 14 February 2001 Close book. Closed notes. No Calculators allowed. Answers available when you leave. Each question is worth 20 points. 1. A number of students tried to improve on the algorithm for computing the value of a pol
Lehigh - CSC - 340
CSc 340Final ExaminationMonday14 May 2001SUGGESTED ANSWERS 1. Below I give the adjacency matrix of a weighted, undirected graph. The nodes are a, b, c, d, ., m. If there is no arc between two nodes, nothing is shown. If there is an arc betwee
Lehigh - CSC - 340
CSc 340 Test 2 SUGGESTED ANSWERS 1. 2. clientsWednesday21 March 2001Write the specifications of an ADT for a Queue of ints. See Fig. 2.16, p. 90 of the text. For the Queue ADT of question 1: a. Write out the implementation of enqueue and dequeu
Lehigh - IE - 406
Introduction to Mathematical Programming IE496 Quiz 1 ReviewDr. Ted RalphsIE496 Quiz 1 Review1Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material you should know. 1.2, 1.3, 1.5 M
Lehigh - IE - 406
Introduction to Mathematical Programming IE496 Quiz 2 ReviewDr. Ted RalphsIE496 Quiz 2 Review1Reading for The Quiz Material covered in detail in lecture Bertsimas 4.1-4.5, 4.8, 5.1-5.5, 6.1-6.3 Material covered briefly in lecture Bertsimas
Charleston Southern - SOCIO - 203
Cultural NormsBy Mariecelle Raguine
Lehigh - IE - 495
IE 495 Final ExamDue Date: May 1, 2003 Do the following problems. You must work alone on this exam. The only reference materials you are allowed to use on the exam are the textbook by Birge and Louveaux, the Stochastic Programming book by Kall and
Lehigh - CSC - 340
CSc 340 Test 3 Wednesday 18 April 2001 -SUGGESTED ANSWERS -1. Show the nine comparisons which would occur when the tournament algorithm is applied to find the 2nd largest entry in the array x[1],.,x[8] = {8, 4, 9, 7, 3, 2, 6, 1}. In practice, of cour
UMBC - PHYS - 121
Lehigh - IE - 170
Department of Industrial and Systems Engineering Spring 2007Algorithms in Systems Engineering(IE 170)Meeting: Monday, Wednesday, Friday 11:10AM-12PM Monday 1-4PM 375 Packard Lab 444 Mohler LabJeff LinderothOffice: 325 Mohler Office hours: Wedn
Lehigh - IE - 170
IE170 Lab #3Kumar Abhishek &amp; Jeff LinderothIE 170 Problem Set #3: Data StructuresDue Date: February 5, 2006. 11AM. You are to complete lab2 (for which you were given an extra week), and also do the following problems:1Updated Problem Set (#
Lehigh - IE - 495
IE 495 Stochastic Programming Problem Sets #5-#7Due Date: April 28, 2003 Do the following problems. If you work alone, you will receive a 10% bonus on your score. These are the final homework sets for the semester. Problems 12 will be Problem Set #
Lehigh - IE - 170
IE170 Lab #10Prof. LinderothIE 170 Problem Set #10: Maximum FlowsDue Date: April 2, 2006. 11AM.1APSP1.1 Problem Run the Floyd-Warshall Algorithm to compute the shortest path from each vertex to every other vertex in the given graph. Be su
Lehigh - IE - 417
IE 417: Advanced Mathematical ProgrammingInstructor: Office: Phone: E-mail: Office Hours: Web page: Course web page: Course meeting time: Description of Course This course will address a number of advanced topics in mathematical programming with par
Lehigh - IE - 495
IE 495 Stochastic Programming Problem Set #1 - Solutions1Random Linear Programs and the Distribution ProblemRecall the random linear program that we saw in class: minimize x1 + x2 subject to 1 x1 + x2 2 x1 + x2 x1 x2 with 1 U [1, 4] and 2
Lehigh - IE - 417
Final Review IE417In the Beginning.In the beginning, Weierstrass's theorem said that a continuous function achieves a minimum on a compact set. Using this, we showed that for a convex set S and y not in the set, there is a unique point in S with m
Lehigh - IE - 406
Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. RalphsThese questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the linear program pictured here, where the feasible
Lehigh - IE - 406
Quiz 2 Sample Questions IE406 Introduction to Mathematical Programming Dr. RalphsThese questions are from previous years and should you give you some idea of what to expect on Quiz 2. 1. WTH Industries is a producer of two products, Whosamajigits a
Lehigh - IE - 406
Sample Final Examination Questions IE406 Introduction to Mathematical Programming Dr. Ralphs1. Consider the following linear programming problem and its optimal final tableau. min -2x1 - x2 + x3 x1 + 2x2 + x3 8 -x1 + x2 - 2x3 4 x 1 , x2 , x3 0 x
Lehigh - PHYSICS - 21
Previous Lecture Magnetic field energy densityLC oscillationsModified Ampere's LawOctober 24, 2007 p.Today Maxwell's Equations start ac circuits (Phasor diagrams) p.Hour Exam ConflictsIf you are taking Acct 151, you should take t
Lehigh - ME - 242
(Fall 2002) Water with weight density 62.4 lbf / ft 3 flows through a linear resistance of R = 400 lbf s / ft 5 to a tank of area A = .624 ft 2 through a very short tube (no inertance). The applied pressure comprises an infinite series of pulses, eac
Lehigh - PHYSICS - 21
Previous Lecture one slit, two slits diffraction pattern from one slit diffraction pattern from circular aperture p.Today Resolution of optical instruments Polarization Information about the final p.Diffraction by a circular aperture
Lehigh - PHYSICS - 21
Physics 21 Fall 2007Solution to HW-221.2 Lightning occurs when there is a flow of electric charge (principally electrons) between the ground and a thundercloud. The maximum rate of charge flow in a lightning bolt is about 20,000 C/s; this lasts f
Lehigh - PHYSICS - 21
Physics 21 Fall 2007Solution to HW-4Flux out of a Cube A point charge of magnitude Q is at the center of a cube with sides of length L. (A) What is the electric flux through each of the six faces of the cube? (B) What if L changes?22.36 A long
Lehigh - MATH - 205
Week 4: (2.1 Rn and Vector Spaces - finish) 2.2 Subspaces/Spanning 2.3 Independence/Bases 2.4 Nullspaces21. Vector addition; scalar multiplication in R2 . Vectors are x = (x, y), with x, y real numbers. (x1 , y1 ) + (x2 , y2 ) = (x1 + x2 , y1 + y
Lehigh - PHYSICS - 31
Physics 31: Homework #5 Due Thursday, February 21, 2008 Problem A: The wave function of a particle is given by (x) = N exp - x 3.1915 A2(When you evaluate this function, take all distances x to be in .) A (a) What is the correct numerical value o
Lehigh - PHYSICS - 21
Physics 21 Fall 2007Visualizing Electric Field LinesSolution to HW-3D21.96 Positive charge Q is uniformly distributed around a semicircle of radius a. Find the components of the electric field at the center of curvature P.(A) For a sheet of n
Lehigh - PHYSICS - 21
Previous Lecture two slits, many slits. The important ratio is /d, which gives angular separation of interference peaks p.Today and Thursday Course evaluation More on diffraction one slit Demonstrations Resolution of optical instruments
Lehigh - PHYSICS - 21
Physics 21 Fall, 2007General InformationGrading: Grades are based on the following scheme: First Exam (Oct. 4) Second Exam (Nov. 1) Five Quizzes Homework (on the web) Recitation leader's assessment Final Exam Total 100 100 50 50 50 200 550Subje
Lehigh - PHYSICS - 21
Previous LectureEM waves satisfy the Wave Equation1 E E - 2 2 =0 2 x c t 2 2 1 B B - 2 2 =0 x2 c t p.22Today Upcoming HW assignments Geometric vs. Physical Optics Mirrors p.HomeworkHW-24 is due tomorrow, Nov. 16, and then HW-25 i
Lehigh - PHYSICS - 21
Previous Lecture Geometric Optics Ray tracing Mirror Equation 1 1 1 + = do di f p.Today Lenses Several demos p.The Mirror Equation1 1 1 + = di do f p.The Mirror Equation1 1 1 + = di do f1 1 1 + = 2f 2f f p.Real vs. Virtua
Lehigh - MATH - 205
Week 3b: 3.6 Cooling/Mixing [see Cooling example, week 3] [3.9 Numerical Solutions [Euler, Maple's dsolve/numeric] not collected 4.1 Higher Order DE 4.2 Constant Coef, Homogeneous DE - Problem: A 200L tank is half full of a solution containing 100g o
Lehigh - PHYSICS - 21
Previous Lecture Refraction (Snell's Law) Ray tracing for lenses Lens Equation: 1 1 1 + = do di f p.Today Rainbows Lensmaker's Equation Two lens systems (e.g., telescope) Aberrations p.Rainbows refraction total internal reflect
Lehigh - MATH - 75 - 76
Mathematics 75 Limit Proofs October 25, 2006 Every time you're asked to prove that lim f (x) = L, your answer will xa follow the same general form. How you fill in some of the details may vary, but the framework remains the same each time. An example
Lehigh - IE - 426
IE 398 Final ExamDecember 11, 2002. 8:00AM-11:00AM There are 180 points total on this exam. Please put your name on all the pages of the exam. If you need more space, feel free to use the backs of the sheets. The more clearly you write your ans
Lehigh - IE - 426
Review and Catchup Nonlinear ProgrammingReview and Catchup Nonlinear ProgrammingStochastic ProgrammingHomework Questions? IE426: Optimization Models and Applications: Lecture 22Jeff LinderothDepartment of Industrial and Systems Engineering Le
Lehigh - PHYSICS - 21
Previous Lecture Magnetic domains in iron Hysteresis Types of magnetism (ferro-, para-, dia-)October 15, 2007 p.Today: Faraday's Law p.Today: Faraday's Lawand related topics: finish Paramagnetism Diamagnetism Lenz' Law p.Parama
Lehigh - PHYSICS - 21
Previous LectureFaraday's Law:d E dl = - dt B dA = - dB dA dtE is related toB . tLenz's Law p.TodayA Big Generalization of Faraday's Law Mutual inductance M Self inductance L p.Example 29.1 p.A Big GeneralizationWe di
Lehigh - PHYSICS - 21
Physics 21 Fall 2007Solution to HW-724.30 A parallel plate capacitor has energy U stored in it. The separation between the plates is d_old. If the separation is decreased to d_new, (A) what is the energy stored if the capacitor is disconnected fr
Lehigh - MATH - 75 - 76
On the assigned problems from 2.5, #1 is similar to problem #43 on page 134. Each part of this problem is similar, so I'll only do part (a). x2 - 2x - 8 #43, pg 143(a) Show that f (x) = has a removable x+2 discontinuity at x = -2 and find a function
Lehigh - MATH - 75 - 76
Math 75 Fall 2006, Lehigh University, Exam I review These are a sampling of problems for review for the first 4o clock test. This sample may not be complete, and is not representative of the test questions. Be sure to prepare from all material from c
Lehigh - MATH - 75 - 76
ASSIGNED PROBLEMS-MATH 75 APPENDIX A 1. Solve the inequality: 2 &lt; |3 - x| 82.Solve the equation: |2x - 3| = |3 - x| 3.Solve the inequality: x2 - 3x - 10 &gt; 0APPENDIX B 1. Find the equation for the line passing through (2,9) with a y-intercept of
Lehigh - MATH - 75 - 76
Calculus IThis document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at http:/tutorial.math.lamar.edu/terms.asp. The online version of this document
Lehigh - IE - 170
Algorithms in Systems Engineering IE170 Lecture 7Dr. Ted RalphsIE170 Lecture 71References for Today's Lecture Required reading CLRS Chapter 6 References D.E. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching (Third Ed
Lehigh - PHYSICS - 21
Previous Lecture Maxwell's Equations Phasors for ac circuitsOctober 29, 2007 p.Today Exam and Schedule Issues More on Phasors Transformers (not on test) p.Hour Exam Info for 95% of youIf you have no conflicts and are not authorized
Lehigh - PHYSICS - 21
Previous Lecture Model for Current Resistors in series and parallel Kirchhoff's Rules p.RC CircuitsAdd capacitors to the circuits. p.Nomenclature Steady-state Time-dependent (transient) p.RC CircuitsSwitch is closed at t = 0, and
Lehigh - PHYSICS - 21
Previous Lecture Ucap = u=1 2 CV 2for capacitorEnergy is stored in electric field:2 1 0E 2 Dielectric constant K Current I is flow of charge p.Current: Realistic ModelSmall drift velocity vd superimposed on large random motion (vr
Lehigh - IE - 170
IE 170 Final Examination Practice ProblemsDr. T.K. Ralphs1. (from David Eppstein, UC Irvine) Suppose you are implementing a spreadsheet program in which you must maintain a grid of cells. Some cells contain values, while other cells contain formul
Lehigh - PHYSICS - 21
Previous LectureV and E are related: V (r) = - E = -VP rE dlCapacitors in series and parallel: Ceff = C1 + C2 parallel 1 1 1 = + series Ceff C1 C2 p.EnergyIt takes work to build up a charge on a capacitor. p.EnergyIt takes work to bui
Lehigh - PHYSICS - 21
Previous Lecture Mutual inductance Self inductanceLR circuitsOctober 22, 2007 p.Hour Exam 2 Thursday, Nov. 1 at 4:10 pm LL270, Whitaker 303, &amp; Williams 301 Covers reading assignments through L-18 (end of Ch. 31) More on the web this we
Lehigh - IE - 426
IE426 Problem Set #1Prof Jeff LinderothIE 426 Problem Set #1Due Date: September 14, 2006. 4:30PM.1Convex FunctionsCategorize each of the functions below as convex, concave, or nonconvex over its domain. Note: Be sure to consider whether o
Lehigh - PHYSICS - 21
Previous LectureWe completed the geometric optics of lenses and mirrors: location of images index of refraction dispersion total internal reflection Lensmaker's Equation one and two lens systems optical aberrations p.Geometric Optics L
Lehigh - PHYSICS - 21
Previous LectureMechanical waves satisfy the Wave Equation1 2D 2D - 2 2 =0 x2 v tNovember 12, 2007 p.Today Energy transport in waves Maxwell's Equations have wave-like solutions. p.Energy transportMechanical waves transport energy
Lehigh - ENGR - 1
math.h fstream.h string.h iostream.h iomanip.h = = equal to &amp; and | or ! not/* setiosflags(ios:type) = sets type of output to use= / resetiosflags -ios:scientific = 2 . 7 4 e 0 0 1 -ios:fixed = 3 . 2 5 setiosflags(ios:location) =sets location = rig
Lehigh - ENGR - 1
int i,j; /&lt;- BUBBLE SORT double temp1,temp2; for(i=0;i&lt;n;i+) /n is the number of memory locations needing to be sorted { for(j=0;j&lt;n-1;j+) { if(array[j]&gt;array[j+1]) { temp1=array[j]; temp2=array[j+1]; array[j]=temp2; array[j+1]=temp1; } } } / finds t
Lehigh - IE - 426
IE426 Problem Set #2 AnswersProf Jeff LinderothIE 426 Problem Set #2 - Answers1LP KnowledgeFor each of the problems in this section, you should determine which of the characterizations below best describes the linear program. The linear p
Lehigh - IE - 426
IE426 Problem Set #1Prof Jeff LinderothIE 426 Problem Set #1 Answers 1 Convex FunctionsCategorize each of the functions below as convex, concave, or nonconvex over its domain. Note: Be sure to consider whether or not each function is in more t
Lehigh - IE - 426
IE426 Problem Set #2Prof Jeff LinderothIE 426 Problem Set #2Due Date: September 26, 2006. 4:30PM. Note: No late homework will be accepted, as I will be discussing solutions on 9/261LP KnowledgeFor each of the problems in this section, you