Linear Algebra Problems - Department of Mathematics

1 Linear Algebra ProblemsMath 504 505 Jerry L. KazdanTopics1 Basics2 Linear Equations3 Linear Maps4 Rank One Matrices5 Algebra of Matrices6 Eigenvalues and Eigenvectors7 Inner Products and Quadratic Forms8 Norms and Metrics9 Projections and Reflections10 Similar Matrices11 Symmetric and Self-adjoint Maps12 Orthogonal and Unitary Maps13 Normal Matrices14 Symplectic Maps15 Differential Equations16 Least Squares17 Markov Chains18 The Exponential Map19 Jordan Form20 Derivatives of Matrices21 Tridiagonal Matrices22 Block Matrices23 Interpolation24 Dependence on Parameters25 Miscellaneous ProblemsThe level of difficulty of these Problems varies wildly. Some are entirely appropriate for ahigh school course.

2 Others definitely Problems are categorized by topics, this should not be taken very seriously. Manyproblems fit equally well in several different :To make this collection more stablenonew Problems will be added in the course corrections and clarifications will be inserted. Corrections and comments arewelcome. Email: have never formally written solutions to these Problems . However, I have frequently usedsome in Homework and Exams in my own Linear Algebra courses in which I often havewritten solutions. See my web page: ~kazdan/Notation:We occasionally writeM(n,F) for the ring of alln nmatrices over the fieldF,whereFis a real matrixAwe sometimes use that the adjointA is thetranspose and Basics1. At noon the minute and hour hands of a clock ) What in the first time,T1, when they are perpendicular?

3 B) What is the next time,T2, when they again coincide?12. Which of the following sets are Linear spaces?a){X= (x1, x2, x3) inR3with the propertyx1 2x3= 0}b) The set of solutions~xofA~x= 0, whereAis anm ) The set of 2 2 matricesAwith det(A) = ) The set of polynomialsp(x) withR1 1p(x)dx= ) The set of solutionsy=y(t) ofy + 4y +y= ) The set of solutionsy=y(t) ofy + 4y +y= ) LetSfbe the set of solutionsu(t) of the differential equationu xu=f(x). Forwhich continuous functionsfisSfa Linear space? Why? [Note:You are notbeing asked to actually solve this differential equation.]3. Which of the following sets of vectors are bases forR2?a).{(0,1),(1,1)}b).{(1,0),(0,1),(1 ,1)}c).{(1,0),( 1,0}d).{(1,1),(1, 1)}e).{((1,1),(2,2)}f).{(1,2)}4. For which real numbersxdo the vectors: (x,1,1,1), (1, x,1,1), (1,1, x,1), (1,1,1, x)notform a basis ofR4?

4 For each of the values ofxthat you find, what is the dimensionof the subspace ofR4that they span?5. LetC(R) be the Linear space of all continuous functions ) LetScbe the set of differentiable functionsu(x) that satisfy the differential equa-tionu = 2xu+cfor all realx. For which value(s) of the real constantcis this set a Linear subspaceofC(R)?b) LetC2(R) be the Linear space of all functions fromRtoRthat have two continuousderivatives and letSfbe the set of solutionsu(x) C2(R) of the differentialequationu +u=f(x)for all realx. For which polynomialsf(x) is the setSfa Linear subspace ofC(R)?c) LetAandBbe Linear spaces andL:A Bbe a Linear map. For which vectorsy Bis the setSy:={x A|Lx=y}a Linear space?26. LetPkbe the space of polynomials of degree at mostkand define the Linear mapL:Pk Pk+1byLp:=p (x) +xp(x).

5 A) Show that the polynomialq(x) = 1 is not in the image ofL. [Suggestion:Trythe casek= 2 first.]b) LetV={q(x) Pk+1|q(0) = 0}. Show that the mapL:Pk Vis invertible.[Again, tryk= 2 first.]7. Compute the dimension and find bases for the following Linear ) Real anti-symmetric 4 4 ) Quartic polynomialspwith the property thatp(2) = 0 andp(3) = ) Cubic polynomialsp(x, y) in two real variables with the properties:p(0,0) = 0,p(1,0) = 0 andp(0,1) = ) The space of Linear mapsL:R5 R3whose kernels contain (0,2, 3,0,1).8. a) Compute the dimension of the intersection of the following two planes inR3x+ 2y z= 0,3x 3y+z= ) A mapL:R3 R2is defined by the matrixL:= 1 2 13 3 1 . Find thenullspace (kernel) IfAis a 5 5 matrix with detA= 1, compute det( 2A).

6 10. Does an 8-dimensional vector space contain Linear subspacesV1,V2,V3with no com-mon non-zero element, such thata). dim(Vi) = 5,i= 1,2,3?b). dim(Vi) = 6,i= 1,2,3?11. LetUandVboth be two-dimensional subspaces ofR5, and letW=U V. Find allpossible values for the dimension LetUandVboth be two-dimensional subspaces ofR5, and define the setW:=U+Vas the set of all vectorsw=u+vwhereu Uandv Vcan be any ) Show thatWis a Linear ) Find all possible values for the dimension LetAbe ann nmatrix of real or complex numbers. Which of the following statementsareequivalentto: the matrixAis invertible ?3a) The columns ofAare linearly ) The columns ) The rows ofAare linearly ) The kernel ofAis ) The only solution of the homogeneous equationsAx= 0 isx= ) The Linear transformationTA:Rn Rndefined byAis ) The Linear transformationTA:Rn Rndefined byAis ) The rank ) The adjoint,A , is ) detA6= Call a subsetSof a vector spaceVaspanning setif Span(S) =V.

7 Suppose thatT:V Wis a Linear map of vector ) Prove that a Linear mapTis 1-1 if and only ifTsends linearly independent setsto linearly independent ) Prove thatTis onto if and only ifTsends spanning sets to spanning Linear Equations15. Solve the given system or show that no solution exists:x+ 2y= 13x+ 2y+ 4z= 7 2x+y 2z= 116. Say you haveklinear algebraic equations innvariables; in matrix form we writeAX=Y. Give a proof or counterexample for each of the ) Ifn=kthere is alwaysat most ) Ifn > kyou canalwayssolveAX= ) Ifn > kthe nullspace ofAhas dimension greater than ) Ifn < kthen forsomeYthere isnosolution ofAX= ) Ifn < ktheonlysolution ofAX= 0 isX= LetA:Rn Rkbe a Linear map. Show that the following are ) For everyy Rkthe equationAx=yhas at most one )Ais injective (hencen k).

8 [injectivemeansone-to-one]c) dim ker(A) = )A is surjective (onto).e) The columns ofAare linearly LetA:Rn Rkbe a Linear map. Show that the following are ) For everyy Rkthe equationAx=yhas at least one )Ais surjective (hencen k). [surjectivemeansonto]c) dim im(A) = )A is injective (one-to-one).e) The columns LetAbe a 4 4 matrix with determinant 7. Give a proof or counterexample for eachof the ) For some vectorbthe equationAx=bhas exactly one ) For some vectorbthe equationAx=bhas infinitely many ) For some vectorbthe equationAx=bhas no ) For all vectorsbthe equationAx=bhas at least one LetA:Rn Rkbe a real matrix, not necessarily ) If two rows ofAare the same, show thatAis not onto by finding a vectory=(y1, .. , yk) that is not in the image ofA.

9 [Hint:This is a mental computation ifyou write out the equationsAx=yexplicitly.]b) What ifA:Cn Ckis a complex matrix?c) More generally, if the rows ofAare linearly dependent, show that it is not LetA:Rn Rkbe a real matrix, not necessarily ) If two columns ofAare the same, show thatAis not one-to-one by finding a vectorx= (x1, .. , xn) that is in the nullspace ) More generally, if the columns ofAare linearly dependent, show thatAis LetAandBben nmatrices withAB= 0. Give a proof or counterexample foreach of the ) EitherA= 0 orB= 0 (or both).b)BA= 0c) If detA= 3, thenB= ) IfBis invertible thenA= ) There is a vectorV6= 0 such thatBAV= Consider the system of equationsx+y z=ax y+ 2z= ) Find the general solution of the homogeneous ) A particular solution of the inhomogeneous equations whena= 1 andb= 2isx= 1, y= 1, z= 1.

10 Find the most general solution of the ) Find some particular solution of the inhomogeneous equations whena= 1 andb= ) Find some particular solution of the inhomogeneous equations whena= 3 andb= 6.[Remark: After you have done part a), it is possible immediately to write the solutionsto the remaining parts.]24. Solve the equations2x+ 3y+ 2z=1x+ 0y+ 3z=22x+ 2y+ 3z=3forx,y, :IfA= 2 3 21 0 32 2 3 ,thenA 1= 6 5 93 2 42 2 3 .25. Consider the system of Linear equationskx+y+z=1x+ky+z=1x+y+kz= what value(s) ofkdoes this have (i) a unique solution? (ii), no solution?(iii) infinitely many solutions? (Justify your assertions).26. LetA= 1 1 11 1 2 .6a) Find the general solutionZof the homogeneous equationAZ= ) Find some solution ofAX= 12 c) Find the general solution of the equation in part b).