Math 54: Linear Algebra and Differential Equations Worksheets

1 Math 54: Linear Algebra andDifferential Equations Worksheets7thEditionDepartment of Mathematics, University of California at BerkeleyiMath 54 Worksheets ,7thEditionPrefaceThis booklet contains the Worksheets for Math 54, Berkeley s Linear Algebra introduction to each worksheet very briefly motivates the main ideas but is notintended as a substitute for the textbooks or lectures. The questions emphasize qualitativeissues and the problems are more computationally intensive. The additional problems aremore challenging and sometimes deal with technical details, or tangential , more problems were provided on each worksheet than can be completed duringa discussion period. This was not a scheme to frustrate the student; rather, we aimed toprovide a variety of problems that can reflect different aspects professors and s maychoose to Stein coordinated the 5th edition in consultation with Tom Insel; then MichaelWu has reorganized the 2000 edition.

2 Michael Hutchings made tiny changes in 2012 for the7th to this workbook include:Michael AuConcha GomezZeph GrunschlagGeorge JohnsonReese JonesDavid KohelBob PrattWilliam SteinAlan WeinsteinZeph GrunschlagWilliam SteinMath 54 Worksheets ,7thEditioniiContentsLinear Algebra1. Introduction to Linear Systems .. 12. Matrices and Gaussian Elimination .. 33. The Algebra of Matrices .. 64. Inverses and Elementary Matrices..95. Transposes and Symmetry .. 126. Vectors .. 157. General Vector Spaces .. 178. Subspaces, Span, and Nullspaces..199. Linear Independence..2210. Basis and Dimension..2411. Fundamental Subspaces and Rank .. 2612. Error Correcting Codes .. 2913. Linear Transformations .. 3214. Inner Products and Least Squares .. 3515. Orthonormal Bases .. 3816. Determinants .. 4117.

3 Eigenvalues and Eigenvectors .. 4518. Diagonalization .. 4719. Symmetric Matrices .. 50 Differential Equations20. The Wronskian and Linear Independence .. 5221. Higher Order Linear ODEs .. 5422. Homogeneous Linear ODEs .. 5723. Systems of First Order Linear Equations ..5924. Systems of First Order Equations Continued..6225. Oscillations of Shock Absorbers .. 65iiiMath 54 Worksheets ,7thEdition26. Introduction to Partial Differential Equations .. 6827. Partial Differential Equations and Fourier Series .. 7028. Applications of Partial Differential Equations .. 721 Math 54 Worksheets , to Linear SystemsIntroductionA Linear equation innvariables is an equation of the forma1x1+a2x2+..+anxn=b,wherea1, a2, .., anandbare real numbers (constants). Notice that a Linear equation doesn tinvolve any roots, products, or powers greater than 1 of the variables, and that there areno logarithmic, exponential, or trigonometric functions of the variables.

4 Solving a linearequation means finding numbersr1, r2, .., rnsuch that the equation is satisfied when wemake the substitutionx1=r1, x2=r2, .. , xn=rn. In this course we will be concernedwith solvingsystemsof Linear Equations , that is, finding a sequence of numbersr1, r2, .. , rnwhich simultaneously satisfy a given set of Equations innvariables. No doubt you havesolved systems of Equations before. In this course we will not only learn techniques forsolving more complicated systems, but we will also be concerning ourselves with importantproperties of the solution sets of systems of (a) What does the graph ofx+ 2y= 5 look like?(b) What does the graph of 2x 3y= 4 look like?(c) Do the two graphs above intersect? If so, what does their intersection look like?2. Write down a system of two Linear Equations in two unknownswhich has no a picture of the Suppose you have a system of two Linear Equations in three unknowns.

5 If a solutionexists, how many are there? What might the set of solutions look like geometrically?Problems1. Solve the following system of Equations and describe in words each step you + 3y z= 13x+ 4y 4z= 73x+ 6y+ 2z= 3 How many solutions are there, and what does the solution set look like geometrically?Math 54 Worksheets ,7thEdition22. Find all solutions of the systemx+y 3z= 5 5x 2y+ 3z= 73x+y z= 3 Describe (but don t draw) the graphs of each of the three above Equations and What condition ona,b,c, anddwill guarantee that there will be exactly one solutionto the following system?ax+by= 1cx+dy= 04. Consider a system of four Equations in three variables. Describe in geometric termsconditions that would correspond to a solution set that(a) is empty.(b) contains a unique point.

6 (c) contains an infinite number of Problems1. Set up a system of Linear Equations for the following problem and then solve it:The three-digit number N is equal to 15 times the sum of its digits. If you reverse thedigits of N, the resulting number is larger by 396. Also, the units (ones) digit of N isone more than the sum of the other two digits. Find Consider the system of equationsax+by=kcx+dy=lex+f y=mShow that if this system has a solution, then at least one equation can be thrown outwithout altering the solution 54 Worksheets , and Gaussian EliminationIntroductionA Linear system corresponds to an augmented matrix, and the operations we use on a linearsystem to solve it correspond to the elementary row operations we use to change a matrixinto row echelon form. The process is calledGaussian elimination, and will come in handyfor the rest of the : The augmented matrix for the system2x1 3x2+x3 x4= 43x1+ 2x2+x3 3x4= 15x1+x2 x3+x4= 32x1 5x2+ 4x3 6x4= 6is 2 3 1 13 2 1 35 1 1 12 5 4 6.

7 2. (a) Identify the first pivot of the matrix 1 1 2 1 2 3 5 0 1 2 1 01 0 1 3 .(b) If that pivot in part (a) was not in the first row, interchange rows so that it is.(c) Now add suitable multiples of the first row to the other rowsto make all otherentries in the first column zero.(d) Ignoring the first row, find the next pivot and repeat steps(b) and (c) on thesecond column.(e) Continue until all of the rows that contain only zeros areat the bottom of thematrix and each pivot appears to the right of all the pivots above You and a friend rent a room in an old house and find that if both of you are using yourblow dryers, the 20 amp fuse for that circuit occasionally blows. Each blow dryer hasa high and a low power setting, which has the effect of fixing theelectrical wire from the fuse box is as old as the house, and has an additional resistancewhich newer wiring would 54 Worksheets ,7thEdition4 Rwiw120Vi1i2R1R2 Equations for the current flowing through each element of thecircuit are obtained fromKirchhoff s laws.

8 The first equation states that the current flowing through the wiregoes into one or the other blow dryer, so thatiw=i1+i2wherewrefers to the wire, while 1 and 2 refer to the blow dryers. Two additionalequations result from the fact that for any loop of the circuit, the voltage dissipatedby resistors must equal the source voltage. Thus,Rwiw+R1i1=VRwiw+R2i2=VwhereRw, R1andR2are the resistances andVis the line voltage (120 Volts).(a) Write this system of three Equations in matrix formAX=B, whereXis acolumn vector whose entries are the three unknown currents.(b) Solve this matrix equation for the currents when both blow dryers are in three cases: both blow dryers operating at low power; both at highpower; and one on low and one on high. Let the two power settings be 1000W and1500W, for which the associated resistances are 15 and 10 ,respectively.

9 (Notethat higher resistance reduces the power drawn by the dryer.) In all cases, letthe wire resistance beRw= . Can both blow dryers be used simultaneouslyunder any conditions? Under what conditions will the fuse blow?2. Write down the augmented matrix for the given system of Equations and then reduceto row echelon form.(a)x1+ 2x2 x4= 1 x1 3x2+x3+ 2x4= 3x1 x2+ 3x3+x4= 12x1 3x2+ 7x3+ 3x4= 4(b)x1+ 2x2 3x3= 92x1 x2+x3= 03x1 2x2+ 4x3= 04x1 x2+x3= 45 Math 54 Worksheets ,7thEdition3. Solve the system in part (a) of problem Suppose you accept a software maintenance job in which youmake $80 a day for eachday you show up to work, but are penalized $20 per day that you don t go to 60 days you find you ve earned $2200. How many days have yougone to work?(Assume that you were expected to work during each of the 60 days.)

10 You may wishto set up a system of two Linear Equations and solve Find a Linear system in 3 variables, or show that none exists, which:(a) has the unique solutionx= 2,y= 3,z= 4.(b) has infinitely many solutions, includingx= 2,y= 3,z= As you know, two points determine a line. But what does this mean? The equation ofa line isax+by+c= 0. Use a Linear system to find an equation of the line throughthe points ( 1,1) and (2,0). Check your answer. How can two Equations determinethe three unknownsa,b, andc?Additional Problems1. For which values of does the system( 3)x+y= 0x+ ( 3)y= 0have more than one solution?2. Suppose that the systema11x1+a12x2+a13x3= 0a21x1+a22x2+a23x3= 0a31x1+a32x2+a33x3= 0has onlyx1=x2=x3= 0 as a solution (thetrivial solution). Then consider thesystem obtained from the given system by replacing the threezeros on the right withthree 1 s.