Transcription of Answers to exercises LINEAR ALGEBRA - Joshua
1 LINEAR ALGEBRAJim HefferonThird editionAnswers to ,R+,Rnreal numbers, positive reals,n-tuples of realsN,Cnatural numbers{0,1,2,..}, complex numbers( ),[ ]open interval, closed interval .. sequence (a list in which order matters)hi,jrowiand columnjentry of matrixHV,W,Uvector spaces~v,~0,~0 Vvector, zero vector, zero vector of a spaceVPn,Mn mspace of degreenpolynomials,n mmatrices[S]span of a set B,D ,~ ,~ basis, basis vectorsEn= ~e1, ..,~en standard basis forRnV =Wisomorphic spacesM Ndirect sum of subspacesh,ghomomorphisms ( LINEAR maps)t,stransformations ( LINEAR maps from a space to itself)RepB(~v), RepB,D(h)representation of a vector, a mapZn morZ,In norIzero matrix, identity matrix|T|determinant of the matrixR(h),N(h)range space, null space of the mapR (h),N (h)generalized range space and null spaceGreek letters with pronounciationcharacternamecharactername alphaAL-fuh nuNEW betaBAY-tuh , xiKSIGH , gammaGAM-muhoomicronOM-uh-CRON , deltaDEL-tuh , piPIE epsilonEP-suh-lon rhoROW zetaZAY-tuh , sigmaSIG-muh etaAY-tuh tauTOW (as in cow)
2 , thetaTHAY-tuh , upsilonOOP-suh-LON iotaeye-OH-tuh , phiFEE, or FI (as in hi) kappaKAP-uh chiKI (as in hi) , lambdaLAM-duh , psiSIGH, or PSIGH muMEW , omegaoh-MAY-guhCapitals shown are the ones that differ from Roman are Answers to the exercises inLinear Algebraby J Hefferon. An answerlabeled here as is for the question numbered 4 from the first chapter, secondsection, and third subsection. The Topics are numbered you have an electronic version of this file then save it in the same directory asthe book. That way, clicking on the question number in the book takes you to itsanswer and clicking on the answer number takes you to the question,I welcome bug reports and comments.
3 Contact information is on the book s HefferonSaint Michael s College, Colchester VT USA2017-Jan-01 ContentsChapter One: LINEAR SystemsSolving LINEAR Systems .. : Gauss s Method .. : Describing the Solution Set .. : General=Particular+Homogeneous .. 18 LINEAR Geometry .. : Vectors in Space .. : Length and Angle Measures .. 29 Reduced Echelon Form .. : Gauss-Jordan Reduction .. : The LINEAR Combination Lemma .. 46 Topic: Computer ALGEBRA Systems .. 51 Topic: Accuracy of Computations .. 55 Topic: Analyzing Networks .. 56 Chapter Two: Vector SpacesDefinition of Vector Space .. : Definition and Examples .. : Subspaces and Spanning Sets .. 70 LINEAR Independence.
4 : Definition and Examples .. 81 Basis and Dimension .. : Basis .. : Dimension .. : Vector Spaces and LINEAR Systems .. : Combining Subspaces .. 119ivLinear ALGEBRA , by HefferonTopic: Fields .. 124 Topic: Crystals .. 125 Topic: Voting Paradoxes .. 127 Topic: Dimensional Analysis .. 129 Chapter Three: Maps Between SpacesIsomorphisms .. : Definition and Examples .. : Dimension Characterizes Isomorphism .. 149 Homomorphisms .. : Definition .. : Range space and Null space .. 163 Computing LINEAR Maps .. : Representing LINEAR Maps with Matrices .. : Any Matrix Represents a LINEAR Map .. 185 Matrix Operations .. : Sums and Scalar Products .. : Matrix Multiplication.
5 : Mechanics of Matrix Multiplication .. : Inverses .. 213 Change of Basis .. : Changing Representations of Vectors .. : Changing Map Representations .. 228 Projection .. : Orthogonal Projection Into a Line .. : Gram-Schmidt Orthogonalization .. : Projection Into a Subspace .. 253 Topic: Line of Best Fit .. 263 Topic: Geometry of LINEAR Maps .. 268 Topic: Magic Squares .. 272 Topic: Markov Chains .. 273 Topic: Orthonormal Matrices .. 283 Chapter Four: DeterminantsDefinition .. : Exploration .. : Properties of Determinants .. : The Permutation Expansion .. : Determinants Exist .. 299 Answers to ExercisesiGeometry of Determinants .. : Determinants as Size Functions.
6 301 Laplace s Formula .. : Laplace s Expansion .. 308 Topic: Cramer s Rule .. 314 Topic: Speed of Calculating Determinants .. 315 Topic: Chi s Method .. 315 Topic: Projective Geometry .. 317 Chapter Five: SimilarityComplex Vector Spaces .. 323 Similarity .. : Definition and Examples .. : Diagonalizability .. : Eigenvalues and Eigenvectors .. 336 Nilpotence .. : Self-Composition .. : Strings .. 349 Jordan Form .. : Polynomials of Maps and Matrices .. : Jordan Canonical Form .. 366 Topic: Method of Powers .. 378 Topic: Stable Populations .. 380 Topic: Page Ranking .. 382 Topic: LINEAR Recurrences .. 383 Topic: Coupled Oscillators .. 386iiLinear ALGEBRA , by HefferonChapter OneChapter One: LinearSystemsSolving LINEAR : Gauss s (a)Gauss s Method (1/2) 1+ 2 2x+3y=13 (5/2)y= 15/2gives that the solution isy=3andx=2.
7 (b)Gauss s Method here 3 1+ 2 1+ 3x z=0y+3z=1y=4 2+ 3 x z=0y+3z=1 3z=3givesx= 1,y=4, andz= a system has a contradictory equation then it has no solution. Otherwise,if there are any variables that are not leading a row then it has infinitely manysolution. In the final case, where there is no contradictory equation and everyvariable leads some row, it has a unique solution.(a)Unique solution(b)Infinitely many solutions(c)Infinitely many solutions(d)No solution(e)Infinitely many solutions(f)Infinitely many solutions2 LINEAR ALGEBRA , by Hefferon(g)No solution(h)Infinitely many solutions(i)No solution(j)Unique (a)Gaussian reduction (1/2) 1+ 2 2x+2y=5 5y= 5/2shows thaty=1/2andx=2is the unique solution.
8 (b)Gauss s Method 1+ 2 x+y=12y=3givesy=3/2andx=1/2as the only solution.(c)Row reduction 1+ 2 x 3y+z=14y+z=13shows, because the variablezis not a leading variable in any row, that there aremany solutions.(d)Row reduction 3 1+ 2 x y=10= 1shows that there is no solution.(e)Gauss s Method 1 4 x+y z=102x 2y+z=0x+z=54y+z=20 2 1+ 2 1+ 3x+y z=10 4y+3z= 20 y+2z= 54y+z=20 (1/4) 2+ 3 2+ 4x+y z=10 4y+3z= 20(5/4)z=04z=0gives the unique solution(x,y,z) = (5,5,0). Answers to Exercises3(f)Here Gauss s Method gives (3/2) 1+ 3 2 1+ 42x+z+w=5y w= 1 (5/2)z (5/2)w= 15/2y w= 1 2+ 4 2x+z+w=5y w= 1 (5/2)z (5/2)w= 15/20=0which shows that there are many (a)Gauss s Methodx+y+z=5x y=0y+2z=7 1+ 2 x+y+z=50 2y 1= 5y+2z=7(3/2) 2+ 3 x+y+z=50 2y 1= 5+ (3/2)z=7/2followed by back-substitution givesx=1,y=1, andz=3.
9 (b)Here Gauss s Method3x+z=7x y+3z=4x+2y 5z= 1 (1/3) 1+ 2 (1/3) 1+ 33x+z=7 y+ (8/3)z=5/32y (16/3)z= (10/3)2 2+ 3 3x+z=7 y+ (8/3)z=5/30=0finds that the variablezdoes not lead a row. There are infinitely many solutions.(c)The stepsx+3y+z=0 x y=2 x+y+2z=8 1+ 2 1+ 3x+3y+z=02y+z=24y+3z=8 2 2+ 3 x+3y+z=02y+z=2z=4give(x,y,z) = ( 1, 1,4). (a)Fromx=1 3ywe get that2(1 3y) +y= 3, givingy=1.(b)Fromx=1 3ywe get that2(1 3y) +2y=0, leading to the conclusionthaty=1 of this method must check any potential solutions by substituting backinto all the ALGEBRA , by the reduction 3 1+ 2 x y=10= 3+kto conclude this system has no solutions ifk6=3and ifk=3then it has infinitelymany solutions.
10 It never has a unique ,y=cos , andz=tan :2x y+3z=34x+2y 2z=106x 3y+z=9 2 1+ 2 3 1+ 32x y+3z=34y 8z=4 8z=0givesz=0,y=1, andx=2. Note that no satisfies that (a)Gauss s Method 3 1+ 2 1+ 3 2 1+ 4x 3y=b110y= 3b1+b210y= b1+b310y= 2b1+b4 2+ 3 2+ 4x 3y=b110y= 3b1+b20=2b1 b2+b30=b1 b2+b4shows that this system is consistent if and only if bothb3= 2b1+b2andb4= b1+b2.(b)Reduction 2 1+ 2 1+ 3x1+2x2+3x3=b1x2 3x3= 2b1+b2 2x2+5x3= b1+b32 2+ 3 x1+2x2+3x3=b1x2 3x3= 2b1+b2 x3= 5b1+2b2+b3shows that each ofb1,b2, andb3can be any real number this system alwayshas a unique system with more unknowns than equationsx+y+z=0x+y+z=1has no For example, the fact that we can have the same reaction in twodifferent flasks shows that twice any solution is another, different, solution (if aphysical reaction occurs then there must be at least one nonzero solution).
