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Linear inversion methods and generalized cross-validation

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Online supplement to“Using generalized cross-validation to select parameters in inversions for regional carbonfluxes”Linear inversion methods and generalized cross-validationN IR Y.K RAKAUER AND T APIO S CHNEIDERCalifornia Institute of TechnologyJ AMES T.R ANDERSONEarth System Science,University of California at IrvineS ETH C.O LSENCalifornia Institute of Technology(31August2004)Here we briefly review regularization methods for the solution of linear ill-posed prob-lems,point out relationships among different regularization methods that are used in inversions for regional carbonfluxes,and show how generalized cross-validation canbe used with different regularization methods.The mathematical developments largely follow Hansen[1998].1.Inverse problem for regional carbonfluxesTo estimate CO2fluxes,one has to estimate a vector x in the linear modelAx=b+ε,(1) where b is a given n×1vector of CO2concentrations at n locations;εis a randomerror with zero mean and with covariance matrix cov(ε)=C b;x is an unknownp×1vector of CO2fluxes into and out of p regions;and A is a given n×p matrix representing a transport operator that relates CO2fluxes to CO2concentrations[e.g., Enting,2002].If the transport operator A is ill-conditioned,as is generally the case when the transport is turbulent so that the effect of regional sources and sinks on CO2concen-trations downstream is smoothed out,the least squares estimate of the CO2fluxes is poorly constrained by the CO2concentrations.In inversions for regional CO2fluxes,more stableflux estimates are usually obtained by minimizing,in place of the least squares object function,a regularized object functionJ=(Ax−b)T C−1b(Ax−b)+λ2(x−x0)T C−1x(x−x0),(2) consisting of the sum of the least squares object function(first term)and a penalty term (second term)that penalizes deviations of the solution x from a given prior estimate1Linear inversion methods and generalized cross-validation2 x0.The covariance matrix C x represents uncertainty about the prior estimate x0.The regularization parameterλindicates the relative weight of the penalty term compared with the least squares term.In CO2inversions,the covariance matrices C b and C x are usually taken to be di-agonal,with diagonal entries c b and c x equal to assumed variances of the local CO2 concentration errors and of the regional priorflux distributions.(However,the meth-ods presented here may be used regardless of whether the covariance matrices are di-agonal.)The regularization parameterλis usually taken to be equal to one.In theTransCom protocol,which we followed,the prior standard deviations c1/2x for land re-gions are taken to be proportional to the growing season net CO2fluxes estimated witha model of the biosphere;the prior standard deviations c1/2x for ocean regions are taken to be proportional to the area of each region and to the number of CO2measurements in each region[Gurney et al.,2003].The minimizer x∗of the object function(2)forλ=1can be interpreted as a Bayesian maximum a posteriori estimate of CO2fluxes,assuming a Gaussian distribu-tion of priorfluxes with mean x0and covariance matrix C x[Tarantola,1987;Enting, 2002].Alternatively,the minimizer x∗of the object function(2)for anyλcan be in-terpreted as a Tikhonov-regularized estimate of CO2fluxes[Tikhonov,1963;Hansen, 1998,chapter5].(Tikhonov regularization is also known as ridge regression[Hoerl and Kennard,1970].)In the Bayesian interpretation,the weighting matrix,or inverse of the prior covariance matrix,C−1x is taken to be known a priori.In the regularization interpretation,the weighting matrix C−1x is taken to be known up to the scaling factor λ,a regularization parameter that must be estimated.2.Transformation to standard formThe object function(2)can be transformed to a standard form by mapping the prior estimate x0to zero and by rescaling variables so that the covariance matrices C b and C x,assumed to be nonsingular,are identity matrices[Hansen,1998,chapter2.3].The transformation takes the form¯A=C−1/2bAC1/2x,(3a)¯x=C−1/2x(x−x0),(3b)¯b=C−1/2b(b−Ax0),(3c)where C1/2b and C1/2x are the Cholesky factors of the covariance matrices C b and C x.The linear model(1)in the original variables is equivalent to the linear model¯A¯x=¯b+¯ε(4) in the transformed variables,with an error covariance matrix cov(¯ε)equal to the iden-tity matrix.In the transformed variables(3),the object function(2)assumes the stan-dard formJ= ¯A¯x−¯b 22+λ2 ¯x 22,(5) where · 2denotes the Euclidean norm.Forλ=0,minimizing the object function(5) yields the least squares estimates.Forλ=1,minimizing the object function(5)yieldsLinear inversion methods and generalized cross-validation3 the Bayesian estimates used in the TransCom CO2inversions[Gurney et al.,2003] as well as in most other CO2inversions starting with Enting[1993].For arbitrary λ>0,minimizing the object function(5)yields the Tikhonov-regularized estimate with regularization parameterλ.3.Singular value decomposition,filter factors,and regularization methods The least squares estimate and several regularized estimates for the linear model(4) can be expressed compactly in terms of the singular value decomposition of the trans-formed transport operator¯A,¯A=UΣV T,(6) where U and V have orthonormal columns u i(left singular vectors)and v i(right singular vectors),andΣis a diagonal matrix with diagonal entriesσi≥0,which are assumed to be arranged in descending order.A large family of estimates¯x∗for the linear model(4)can be expressed as a linear combination of right singular vectors v i,¯x∗=rank(¯A)i=1f iu T i¯bσiv i,(7)where thefilter factors f i characterize the estimation method[cf.Hansen,1998,chap-ter4].The coefficients u T i¯b are often referred to as Fourier coefficients,in analogy to inverse problems in which the counterpart of the matrix¯A is a convolution operator whose singular value decomposition is equivalent to a Fourier expansion[cf.Wahba, 1977;Anderssen and Prenter,1981].a.Least squares estimationFor the least squares estimate(λ=0),thefilter factors are identically equal to one (that is,nofiltering),f i=1for all i.(8) Expressing the least squares estimate in terms of the singular value decomposition(7) makes manifest that errors of orderεin the transformed data¯b typically result in errors of orderε/σmin in the estimate¯x∗,whereσmin is the smallest nonzero singular value. If typical data errors exceed the smallest singular value,the least squares estimate is poorly constrained by the data.If the transformed transport operator¯A is rank-deficient(i.e.,rank(¯A)<p),the least squares estimate is not unique.In this case,the estimate(7)withfilter factors(8)is the least squares estimate with minimum norm ¯x∗ 2.If the transformed transport operator¯A has small singular values,regularization methods stabilize the least squares estimates byfiltering out the contributions of right singular vectors v i that are associated with the small singular valuesσi.These contri-butions to the estimate(7)typically represent high-frequency noise that is not estimable given the uncertainty of the data.Linear inversion methods and generalized cross-validation4b.Bayesian estimationFor the Bayesian maximum a posteriori estimate(λ=1),for which a prior normal dis-tribution with mean zero and identity covariance matrix is assumed for the transformedfluxes¯x,thefilter factors aref i=σ2iσ2i+1.(9)Thisfilter function decays smoothly from f i≈1forσi 1to f i≈0forσi 1;that is,right singular vectors with singular values smaller than1are effectivelyfiltered out. Thisfiltering is what is commonly used in inversions for CO2fluxes.c.Tikhonov regularization/ridge regressionFor the Tikhonov-regularized estimate(λadjustable),thefilter factors are[Hansen,1998,chapter4.2]f i=σ2iσ2i+λ2.(10)Thisfilter function decays smoothly from f i≈1forσi λto f i≈0forσi λ; that is,right singular vectors with singular values smaller thanλare effectivelyfiltered out.The Tikhonovfilter function is structurally identical to the Wienerfilter,which is the optimalfilter to separate noise of spectral densityλ2from a signal of spectral densityσ2i[Papoulis,1991;Anderssen and Prenter,1981].d.Least squares estimation with inequality constraintsThe estimate(7)with Tikhonovfilter factors(10)is also the solution of a least squares problem with inequality constraint,min¯x¯A¯x−¯b 22subject to ¯x 22≤α,(11)whereαis a parameter constraining the norm of the solution.If the norm ¯x∗ 2of the least squares estimate is less thanα,the least squares estimate solves the constrained least squares problem(11).If the norm of the least squares estimate is greater than α,the Tikhonov estimate solves the constrained least squares problem(11),with a regularization parameterλ(a Lagrange multiplier)that is a function ofα[Golub and Van Loan,1989,chapter12.1.2].Regularization with an inequality constraint(11),then,is equivalent to Tikhonov regularization if the inequality constraint is not redundant.Bayesian estimation and regularization with an inequality constraint,contrasted by Fan et al.[1999]as differ-ent methods,are therefore very similar.The methods merely correspond to choosing different values of the regularization parameterλ(and potentially different scalings of the variables).Linear inversion methods and generalized cross-validation5e.Regularization by truncated singular value decompositionAnother common way tofilter out right singular vectors that are associated with small singular values is to keep only thefirst k right singular vectors,corresponding tofilter-ing with a step functionfilterf i=1i≤k0i>k(12)for some effective rank k≤rank(¯A)[e.g.,Hansen,1998,chapter3.2;Fan et al., 1999].This usually yields estimates similar to Tikhonov regularization with regular-ization parameterλ≈σk.4.Generalized cross-validationGeneralized cross-validation offers a way to estimate appropriate values of parameters such as the regularization parameters k in truncated singular value decomposition,λin Tikhonov regularization,orαin least squares estimation with inequality constraint.In the Bayesian formulation used in TransCom,components of the covariance matrices C b and C x,which are generally poorly known,can likewise be estimated by general-ized cross-validation.For the family of estimates(7),the generalized cross-validation function,to be minimized as a function of the parameters,is given byGCV= ¯A¯x∗−¯b 22T2,(13)where the numerator is the squared residual norm and the denominator is a squared effective number of degrees of freedom[Hansen,1998,chapter7.4].For all estimation methods discussed above,the effective number of degrees of freedom(which is not necessarily an integer)can be written in terms of thefilter factors asT=n−rank(¯A)i=1f i.(14)The residual norm in the numerator of the GCV function can be computed efficiently from a singular value decomposition of the transformed transport operator¯A,making the evaluation of the GCV function for several regularization parameters straightfor-ward.The minimizer of the GCV function approximately minimizes the expected mean squared error of predictions of the transformed data¯b with an estimated linear model (4)[Golub et al.,1979].With small but nonzero probability,the GCV function has a minimum near zero regularization(i.e.,atλ=0or forα→∞),so that generalized cross-validation occasionally leads to undersmoothed estimates when,in fact,more strongly regularized and smoother estimates would be more appropriate[Wahba and Wang,1995].Undersmoothed estimates in such cases can be avoided by constructing bounds for the regularization parameters,for example,from a priori guesses of the magnitude of the residuals[Hansen,1998,chapters7.7and7.2].Linear inversion methods and generalized cross-validation6 In our analyses,we evaluated the GCV function(13)as a function of the regular-ization parameterλand of the weighting parameterτon a mesh with spacing of0.05 inτand of0.27inλ2.Where inversion results are sensitively dependent on inversion parameters,it may be useful not only to choose”optimal”values of the parameters but also to estimate confidence regions for the parameters.Methods that treat inversion parameters as ran-dom variables and estimate their probability distributions given the data and a probabil-ity model for the parameters[Wang and Wahba,1995;Koch,1999;Koch and Kusche, 2002]could be applied for this purpose.Heuristic estimates of confidence regions may also be obtained from the curvature of the GCV function or other object functions at the optimum,by analogy with ordinary least squares regression[Press et al.,1992,chapter 15.6].Given confidence regions,the impact the uncertainty about inversion parameters has onflux estimates could then be quantified using either linear error propagation or Monte-Carlo methods.ReferencesAnderssen,R.S.,and P.M.Prenter(1981),A formal comparison of methods proposed for the numerical solution offirst kind integral equations,J.Austral.Math.Soc.B, 22,488–500.Enting,I.G.(2002),Inverse Problems in Atmospheric Constituent Transport,392pp., Cambridge University Press,Cambridge and New York.Enting,I.G.(1993),Inverse problems in atmospheric constituent studies.3.Estimating errors in surface sources,Inverse Problems,9,649–665.Fan,S.M.,J.L.Sarmiento,M.Gloor,and S.W.Pacala(1999),On the use of reg-ularization techniques in the inverse modeling of atmospheric carbon dioxide,J.Geophys.Res.,104(D17),21503–21512.Golub,G.H.,and C.F.Van Loan(1989),Matrix Computations,2nd ed.,642pp.,Johns Hopkins University Press,Baltimore and London.Golub,G.H.,M.Heath,and G.Wahba(1979),Generalized cross-validation as a method for choosing a good ridge parameter,Technometrics,21,215–223. Gurney,K.R.,w,A.S.Denning,P.J.Rayner,D.Baker,P.Bousquet,L.Bruh-wiler,Y.H.Chen,P.Ciais,S.M.Fan,I.Y.Fung,M.Gloor,M.Heimann,K.Higuchi, J.John,E.Kowalczyk,T.Maki,S.Maksyutov,P.Peylin,M.Prather,B.C.Pak, J.Sarmiento,S.Taguchi,T.Takahashi,and C.W.Yuen(2003),TransCom3CO2 inversion intercomparison:1.Annual mean control results and sensitivity to trans-port and priorflux information,Tellus,55(B2),555–579.Hansen,P.C.(1998),Rank-Deficient and Discrete Ill-Posed Problems:Numerical As-pects of Linear Inversion,247pp.,Society for Industrial and Applied Mathematics, Philadelphia.Hoerl,A.E.,and R.W.Kennard(1970),Ridge regression:Biased estimation for non-orthogonal problems,Technometrics,12,55–67.Koch,K.-R.(1999),Parameter Estimation and Hypothesis Testing in Linear Models, Springer,Berlin,New York.Linear inversion methods and generalized cross-validation7 Koch,K.-R.and J.Kusche(2002),Regularization of geopotential determination fromsatellite data by variance components,J.Geodesy,76(5),259-268.Papoulis,A.(1991),Probability,Random Variables,and Stochastic Processes,3rd ed., 666pp.,McGraw Hill,New York.Press,W.H.,S.A.Teukolsky,W.T.Vetterling and B.P.Flannery(1992),Numeri-cal Recipes in C:The Art of Scientific Computing,994pp.,Cambridge University Press,Cambridge,New York.Tarantola,A.(1987),Inverse Problem Theory:Methods for Data Fitting and Model Parameter Estimation,613pp.,Elsevier,New York.Tikhonov,A.N.(1963),Solution of incorrectly formulated problems and the regu-larization method,Soviet Math.Dokl.,4,1035–1038.English transl.of Doklady Akademii Nauk SSSR,151,501–504.Wahba,G.(1977),Practical approximate solutions to linear operator equations when the data are noisy,SIAM J.Numer.Anal.,14,651–667.Wahba,G.,and Y.Wang(1995),Behavior near zero of the distribution of GCV smooth-ing parameter estimates,Stat.Probabil.Lett.,25,105–111.Wang,Y.D.,and G.Wahba(1995),Bootstrap confidence-intervals for smoothing splines and their comparison to Bayesian confidence-intervals,put.Simul.,51(2-4),263–279.。

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线性不等式组的一种新算法

线性不等式组的一种新算法

Abstract A pivoting-based algorithm for the system of linear inequalities is proposed. Since it solves the system directly without adding any variables, a compact form is used for operations. It not only requires less computation for each iteration but also makes easy the theoretical analysis for the characteristics of the solved problem. It is proved that this method terminates as long as the entering and leaving variables are selected by the smallest-subscript rule in each iteration. This proof is essentially that proposed by G.B. Bland, but varies greatly in the form. The experiment shows that this method is very efficient for solving Markowitz’s portfolio selection model (convex quadratic programming).
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PDF分享：国外优秀数学教材选评

PDF分享：国外优秀数学教材选评

PDF分享：国外优秀数学教材选评《国外优秀数学教材选评》推荐书⽬下载具体内容请查看原：或者注：1. 由于国外教材更新频繁，其中⾮数学专业教材尤为显著，相较于原版本，其中有部分版本已经更新，以下提供的是尽所能搜索到的最新版本2. 所有⽂件都来源⽹络，余不敢独享，特整理发布。

尽⼒择⾼质量版本分享，但难免有所疏漏，烦请提出宝贵意见3. 由于多数书籍出版已多年，购买确有不便，PDF实为⽆奈之举，望有能之⼠可以⽀持正版4. ⽂件为公开分享，链接失效请评论或联系，为了防⽌分享被封禁，添加了密码压缩，密码为：5. ⼗年之前的书籍许多使⽤了DjVu格式+OCR，其质量略⾼于对应扫描PDF，也可以转化为PDF2.2.1 微积分1. Calculus | Hughes Hallet,Gleason,McCallum et al. |2. Calculus : Early Transcendentals | James Stewart | /3. Advanced Calculus | Patrick M. Fitzpatrick |4. Calculus : Early Transcendental functions | Ron Larson, Robert P Hostetler, Bruce H Edwards |5. Calculus One And Sevel Variables | Saturnino L Salas, Einar Hille, Garret J Etgen |6. Calculus | Monty J Strauss, Gerald L Bradley, Karl J Smith |7. Calculus | Dale E Varberg, Edwin J Purcell, Steven E Rigdon |2.2 线性代数1. Linear Algebra and its Applications | Gilbert Strang |2. Introduction to Linear Algebra | Gilbert Strang |3. Linear Algebra and its Applications | David C. Lay |4. Elementary Linear Algebra | Howard Anton | 同下5. Elementary Linear Algebra (application version) | Howard Anton |6. Linear Algebra with Applications | Otto Bretscher |7. Linear Algebra with Applications | Steven J. Leon |8. Linear Algebra Done Right | S. Axler | /2.3 其它1. Differential Equations | P. Blanchard, R. Devaney, G. Hall |2. Concrete Mathematics | R.Graham, D.Knuth, O.Patashnik |3. Discrete Mathematics | Dossey,Otto,Spence,Vanden Eynden |3.1. Introducton to Analysis | Arthur Mattuck |2. Mathematical Analysis | Tom M. Apostol |3. Principle of Mathematical Analysis | Walter Rudin |4. Advanced Calculus | Patrick M. Fitzpatrick |5. Advanced Calculus | Wilfred Kaplan |6. Advanced Calculus | Lynn H. Loomis, Shlomo Sternberg |7. Problems and Theorems in Analysis | George Po l ya, Ga b or Szego|8. Functional Analysis | Walter Rudin |4.1. Complex Analysis | Lars V. Ahlfors |2. Introduction to complex analysis | Kunihiko Kodaira |3. Functions of One Complex Variable | John B. Conway |4. Complex Analysis | Serge Lang |5.1. An Introduction to Complex Analysis in Several Variables | Lars Hörmander |2. Introduction to Complex Analysis Part II, Functions in Several Variables | B. V. Shabat |3. Topics in Complex Function Theory | Carl L. Siegel |6.1. Linear Algebra | K. Hoffmann and R. Kunze |2. Lectures on Linear Algebra | Gelfand |3. Linear Algebra Gems | David Carlson, Charles R. Johnson,David C. Lay, A. Duane Porter |4. Algebra | Michael Artin |5. Codes and Curves | Judy Walker |6. Introduction to Commutative Algebra | Michael Atiyah & I.G.MacDonald |7. Hopf Algebra | Moss E. Sweedler |7.1. Elementary Methods in Number Theory | Melvyn B.Nathanson |2. A Course in Arithmetic | J.-P. Serre |3. Introduction to Analytic Number Theory | Tom Apostol |8.1. An Invitation to Algebraic Geometry | K.Smith etc. |2. Introduction to Commutative Algebra and Algebraic Geometry | Ernst Kunz |3. Basic Algebraic Geometry | Shafarevich |4. Algebraic Geometry | Robin Hartshorne |5. Principles of Algebraic Geometry | Phillip Griffiths, Joseph Harris |6. The Red Book of Varieties and Schemes | David Mumford |7. Compact Complex Surfaces | W.P. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven |9.1. Elements of Algebraic Topology | James R. Munkres |2. Lecture Notes on Elementary Topology and Geometry | Singer & Thorpe |3. Topology from the differentiable viewpoint | John Milnor |4. Algebraic topology | Hatcher |5. Differential forms in algebraic topology | Bott & Tu |6. Knot thoery | Livingston |7. Riemannian Geometry | M.P. Do Carmo |8. Foundations of Differential Geometry (in two volumes) | Shoshichi Kobayashi & Katsumi Nomizu |9. Introduction to Lie groups and Lie algebras | A.A.Sagle & R.E.Walde |10.1. Hyperbolic Partial Differential Equations | Peter D. Lax |2. Partial Differential Equations, An Introduction | Walter A. Strauss |3. Partial Differential Equations | Lawrence C. Evans |4. Partial Differential Equations | Fritz John |11.1. An Introduction to probability theory and its applications, Vol 1 | William Feller |2. A course in Probability Theory | Kai Lai Chung |12.1. Numerical Optimization | J. Nocedal & S. Wright |13.1. Berkeley Problems in Mathematics | Souza & Silva |2. Putnam and Beyond | Gelca & Andreescu |另外推荐：。

线性代数 英文讲义

线性代数 英文讲义

Chapter 1 Matrices and Systems of Equations e Gaussian Elimination method to solve the following linear systems:(1)123423412423423443331733 x x x xx x xx x xx x x-+-=⎧⎪-+=-⎪⎨+-=⎪⎪-++=-⎩(2)1234123412341234 2353 34232 288 7980x x x xx x x xx x x xx x x x+++=⎧⎪+++=-⎪⎨++-=⎪⎪+++=⎩(3) 12341234123442303520320x x x x x x x x x x x x ++-=⎧⎪---+=⎨⎪-++=⎩2.For what value of a that make the linear system consistent ？and solve it. 12341341234123412347382223332112222x x x x x x x x x x x a x x x x x x x x a+++=-⎧⎪+-=⎪⎪+-+=+⎨⎪+++=-⎪⎪+++=⎩3. For what value of c,d that make the linear system consistent ？and solve it. 1234512345234512345132322635433x x x x x x x x x x c x x x x x x x x x d++++=⎧⎪+++-=⎪⎨+++=⎪⎪+++-=⎩4.Suppose 1231231231320352037(1)0x x x x x x x x x x x λλ+-=⎧⎪++=⎪⎨+++=⎪⎪-+=⎩，for what value of λ will the system have infinitely many solutions; for what value of λ will the system have unique solution.5. Suppose A＝1321⎡⎤⎢⎥-⎣⎦，B＝3012⎡⎤⎢⎥⎣⎦，then calculate 23A B-，AB BA-，22A B+。

Linear Algebra and its Applications

Linear Algebra and its Applications

Linear Algebra and its Applications432(2010)2089–2099Contents lists available at ScienceDirect Linear Algebra and its Applications j o u r n a l h o m e p a g e:w w w.e l s e v i e r.c o m/l o c a t e/l aaIntegrating learning theories and application-based modules in teaching linear algebraୋWilliam Martin a,∗,Sergio Loch b,Laurel Cooley c,Scott Dexter d,Draga Vidakovic ea Department of Mathematics and School of Education,210F Family Life Center,NDSU Department#2625,P.O.Box6050,Fargo ND 58105-6050,United Statesb Department of Mathematics,Grand View University,1200Grandview Avenue,Des Moines,IA50316,United Statesc Department of Mathematics,CUNY Graduate Center and Brooklyn College,2900Bedford Avenue,Brooklyn,New York11210, United Statesd Department of Computer and Information Science,CUNY Brooklyn College,2900Bedford Avenue Brooklyn,NY11210,United Statese Department of Mathematics and Statistics,Georgia State University,University Plaza,Atlanta,GA30303,United StatesA R T I C L E I N F O AB S T R AC TArticle history:Received2October2008Accepted29August2009Available online30September2009 Submitted by L.Verde-StarAMS classification:Primary:97H60Secondary:97C30Keywords:Linear algebraLearning theoryCurriculumPedagogyConstructivist theoriesAPOS–Action–Process–Object–Schema Theoretical frameworkEncapsulated process The research team of The Linear Algebra Project developed and implemented a curriculum and a pedagogy for parallel courses in (a)linear algebra and(b)learning theory as applied to the study of mathematics with an emphasis on linear algebra.The purpose of the ongoing research,partially funded by the National Science Foundation,is to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of individuals in both domains.The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted,in some students,a rich understanding of both domains and that had a mutually reinforcing effect.Furthermore,there is evidence that the deeper insights will contribute to more effective instruction by those who become high school math teachers and,consequently,better learning by their students.The courses developed were appropriate for mathematics majors,pre-service secondary mathematics teachers, and practicing mathematics teachers.The learning seminar focused most heavily on constructivist theories,although it also examinedThe work reported in this paper was partially supported by funding from the National Science Foundation(DUE CCLI 0442574).∗Corresponding author.Address:NDSU School of Education,NDSU Department of Mathematics,210F Family Life Center, NDSU Department#2625,P.O.Box6050,Fargo ND58105-6050,United States.Tel.:+17012317104;fax:+17012317416.E-mail addresses:william.martin@(W.Martin),sloch@(S.Loch),LCooley@ (L.Cooley),SDexter@(S.Dexter),dvidakovic@(D.Vidakovic).0024-3795/$-see front matter©2009Elsevier Inc.All rights reserved.doi:10.1016/a.2009.08.0302090W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099Thematicized schema Triad–intraInterTransGenetic decomposition Vector additionMatrixMatrix multiplication Matrix representation BasisColumn spaceRow spaceNull space Eigenspace Transformation socio-cultural and historical perspectives.A particular theory, Action–Process–Object–Schema(APOS)[10],was emphasized and examined through the lens of studying linear algebra.APOS has been used in a variety of studies focusing on student understanding of undergraduate mathematics.The linear algebra courses include the standard set of undergraduate topics.This paper reports the re-sults of the learning theory seminar and its effects on students who were simultaneously enrolled in linear algebra and students who had previously completed linear algebra and outlines how prior research has influenced the future direction of the project.©2009Elsevier Inc.All rights reserved.1.Research rationaleThe research team of the Linear Algebra Project(LAP)developed and implemented a curriculum and a pedagogy for parallel courses in linear algebra and learning theory as applied to the study of math-ematics with an emphasis on linear algebra.The purpose of the research,which was partially funded by the National Science Foundation(DUE CCLI0442574),was to investigate how the parallel study of learning theories and advanced mathematics influences the development of thinking of high school mathematics teachers,in both domains.The researchers found that the particular synergy afforded by the parallel study of math and learning theory promoted,in some teachers,a richer understanding of both domains that had a mutually reinforcing effect and affected their thinking about their identities and practices as teachers.It has been observed that linear algebra courses often are viewed by students as a collection of definitions and procedures to be learned by rote.Scanning the table of contents of many commonly used undergraduate textbooks will provide a common list of terms such as listed here(based on linear algebra texts by Strang[1]and Lang[2]).Vector space Kernel GaussianIndependence Image TriangularLinear combination Inverse Gram–SchmidtSpan Transpose EigenvectorBasis Orthogonal Singular valueSubspace Operator DecompositionProjection Diagonalization LU formMatrix Normal form NormDimension Eignvalue ConditionLinear transformation Similarity IsomorphismRank Diagonalize DeterminantThis is not something unique to linear algebra–a similar situation holds for many undergraduate mathematics courses.Certainly the authors of undergraduate texts do not share this student view of mathematics.In fact,the variety ways in which different authors organize their texts reflects the individual ways in which they have conceptualized introductory linear algebra courses.The wide vari-ability that can be seen in a perusal of the many linear algebra texts that are used is a reflection the many ways that mathematicians think about linear algebra and their beliefs about how students can come to make sense of the content.Instruction in a course is based on considerations of content,pedagogy, resources(texts and other materials),and beliefs about teaching and learning of mathematics.The interplay of these ideas shaped our research project.We deliberately mention two authors with clearly differing perspectives on an undergraduate linear algebra course:Strang’s organization of the material takes an applied or application perspective,while Lang views the material from more of a“pure mathematics”perspective.A review of the wide variety of textbooks to classify and categorize the different views of the subject would reveal a broad variety of perspectives on the teaching of the subject.We have taken a view that seeks to go beyond the mathe-matical content to integrate current theoretical perspectives on the teaching and learning of undergrad-uate mathematics.Our project used integration of mathematical content,applications,and learningW.Martin et al./Linear Algebra and its Applications432(2010)2089–20992091 theories to provide enhanced learning experiences using rich content,student meta cognition,and their own experience and intuition.The project also used co-teaching and collaboration among faculty with expertise in a variety of areas including mathematics,computer science and mathematics education.If one moves beyond the organization of the content of textbooks wefind that at their heart they do cover a common core of the key ideas of linear algebra–all including fundamental concepts such as vector space and linear transformation.These observations lead to our key question“How is one to think about this task of organizing instruction to optimize learning?”In our work we focus on the conception of linear algebra that is developed by the student and its relationship with what we reveal about our own understanding of the subject.It seems that even in cases where researchers consciously study the teaching and learning of linear algebra(or other mathematics topics)the questions are“What does it mean to understand linear algebra?”and“How do I organize instruction so that students develop that conception as fully as possible?”In broadest terms, our work involves(a)simultaneous study of linear algebra and learning theories,(b)having students connect learning theories to their study of linear algebra,and(c)the use of parallel mathematics and education courses and integrated workshops.As students simultaneously study mathematics and learning theory related to the study of mathe-matics,we expect that reflection or meta cognition on their own learning will enable them to construct deeper and more meaningful understanding in both domains.We chose linear algebra for several reasons:It has not been the focus of as much instructional research as calculus,it involves abstraction and proof,and it is taken by many students in different programs for a variety of reasons.It seems to us to involve important mathematical content along with rich applications,with abstraction that builds on experience and intuition.In our pilot study we taught parallel courses:The regular upper division undergraduate linear algebra course and a seminar in learning theories in mathematics education.Early in the project we also organized an intensive three-day workshop for teachers and prospective teachers that included topics in linear algebra and examination of learning theory.In each case(two sets of parallel courses and the workshop)we had students reflect on their learning of linear algebra content and asked them to use their own learning experiences to reflect on the ideas about teaching and learning of mathematics.Students read articles–in the case of the workshop,this reading was in advance of the long weekend session–drawn from mathematics education sources including[3–10].APOS(Action,Process,Object,Schema)is a theoretical framework that has been used by many researchers who study the learning of undergraduate and graduate mathematics[10,11].We include a sketch of the structure of this framework and refer the reader to the literature for more detailed descriptions.More detailed and specific illustrations of its use are widely available[12].The APOS Theoretical Framework involves four levels of understanding that can be described for a wide variety of mathematical concepts such as function,vector space,linear transformation:Action,Process,Object (either an encapsulated process or a thematicized schema),Schema(Intra,inter,trans–triad stages of schema formation).Genetic decomposition is the analysis of a particular concept in which developing understanding is described as a dynamic process of mental constructions that continually develop, abstract,and enrich the structural organization of an individual’s knowledge.We believe that students’simultaneous study of linear algebra along with theoretical examination of teaching and learning–particularly on what it means to develop conceptual understanding in a domain –will promote learning and understanding in both domains.Fundamentally,this reflects our view that conceptual understanding in any domain involves rich mental connections that link important ideas or facts,increasing the individual’s ability to relate new situations and problems to that existing cognitive framework.This view of conceptual understanding of mathematics has been described by various prominent math education researchers such as Hiebert and Carpenter[6]and Hiebert and Lefevre[7].2.Action–Process–Object–Schema theory(APOS)APOS theory is a theoretical perspective of learning based on an interpretation of Piaget’s construc-tivism and poses descriptions of mental constructions that may occur in understanding a mathematical concept.These constructions are called Actions,Processes,Objects,and Schema.2092W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099 An action is a transformation of a mathematical object according to an explicit algorithm seen as externally driven.It may be a manipulation of objects or acting upon a memorized fact.When one reflects upon an action,constructing an internal operation for a transformation,the action begins to be interiorized.A process is this internal transformation of an object.Each step may be described or reflected upon without actually performing it.Processes may be transformed through reversal or coordination with other processes.There are two ways in which an individual may construct an object.A person may reflect on actions applied to a particular process and become aware of the process as a totality.One realizes that transformations(whether actions or processes)can act on the process,and is able to actually construct such transformations.At this point,the individual has reconstructed a process as a cognitive object. In this case we say that the process has been encapsulated into an object.One may also construct a cognitive object by reflecting on a schema,becoming aware of it as a totality.Thus,he or she is able to perform actions on it and we say the individual has thematized the schema into an object.With an object conception one is able to de-encapsulate that object back into the process from which it came, or,in the case of a thematized schema,unpack it into its various components.Piaget and Garcia[13] indicate that thematization has occurred when there is a change from usage or implicit application to consequent use and conceptualization.A schema is a collection of actions,processes,objects,and other previously constructed schemata which are coordinated and synthesized to form mathematical structures utilized in problem situations. Objects may be transformed by higher-level actions,leading to new processes,objects,and schemata. Hence,reconstruction continues in evolving schemata.To illustrate different conceptions of the APOS theory,imagine the following’teaching’scenario.We give students multi-part activities in a technology supported environment.In particular,we assume students are using Maple in the computer lab.The multi-part activities,focusing on vectors and operations,in Maple begin with a given Maple code and drawing.In case of scalar multiplication of the vector,students are asked to substitute one parameter in the Maple code,execute the code and observe what has happened.They are asked to repeat this activity with a different value of the parameter.Then students are asked to predict what will happen in a more general case and to explain their reasoning.Similarly,students may explore addition and subtraction of vectors.In the next part of activity students might be asked to investigate about the commutative property of vector addition.Based on APOS theory,in thefirst part of the activity–in which students are asked to perform certain operation and make observations–our intention is to induce each student’s action conception of that concept.By asking students to imagine what will happen if they make a certain change–but do not physically perform that change–we are hoping to induce a somewhat higher level of students’thinking, the process level.In order to predict what will happen students would have to imagine performing the action based on the actions they performed before(reflective abstraction).Activities designed to explore on vector addition properties require students to encapsulate the process of addition of two vectors into an object on which some other action could be performed.For example,in order for a student to conclude that u+v=v+u,he/she must encapsulate a process of adding two vectors u+v into an object(resulting vector)which can further be compared[action]with another vector representing the addition of v+u.As with all theories of learning,APOS has a limitation that researchers may only observe externally what one produces and discusses.While schemata are viewed as dynamic,the task is to attempt to take a snap shot of understanding at a point in time using a genetic decomposition.A genetic decomposition is a description by the researchers of specific mental constructions one may make in understanding a mathematical concept.As with most theories(economics,physics)that have restrictions,it can still be very useful in describing what is observed.3.Initial researchIn our preliminary study we investigated three research questions:•Do participants make connections between linear algebra content and learning theories?•Do participants reflect upon their own learning in terms of studied learning theories?W.Martin et al./Linear Algebra and its Applications432(2010)2089–20992093•Do participants connect their study of linear algebra and learning theories to the mathematics content or pedagogy for their mathematics teaching?In addition to linear algebra course activities designed to engage students in explorations of concepts and discussions about learning theories and connections between the two domains,we had students construct concept maps and describe how they viewed the connections between the two subjects. We found that some participants saw significant connections and were able to apply APOS theory appropriately to their learning of linear algebra.For example,here is a sketch outline of how one participant described the elements of the APOS framework late in the semester.The student showed a reasonable understanding of the theoretical framework and then was able to provide an example from linear algebra to illustrate the model.The student’s description of the elements of APOS:Action:“Students’approach is to apply‘external’rules tofind solutions.The rules are said to be external because students do not have an internalized understanding of the concept or the procedure tofind a solution.”Process:“At the process level,students are able to solve problems using an internalized understand-ing of the algorithm.They do not need to write out an equation or draw a graph of a function,for example.They can look at a problem and understand what is going on and what the solution might look like.”Object level as performing actions on a process:“At the object level,students have an integrated understanding of the processes used to solve problems relating to a particular concept.They un-derstand how a process can be transformed by different actions.They understand how different processes,with regard to a particular mathematical concept,are related.If a problem does not conform to their particular action-level understanding,they can modify the procedures necessary tofind a solution.”Schema as a‘set’of knowledge that may be modified:“Schema–At the schema level,students possess a set of knowledge related to a particular concept.They are able to modify this set of knowledge as they gain more experience working with the concept and solving different kinds of problems.They see how the concept is related to other concepts and how processes within the concept relate to each other.”She used the ideas of determinant and basis to illustrate her understanding of the framework. (Another student also described how student recognition of the recursive relationship of computations of determinants of different orders corresponded to differing levels of understanding in the APOS framework.)Action conception of determinant:“A student at the action level can use an algorithm to calculate the determinant of a matrix.At this level(at least for me),the formula was complicated enough that I would always check that the determinant was correct byfinding the inverse and multiplying by the original matrix to check the solution.”Process conception of determinant:“The student knows different methods to use to calculate a determinant and can,in some cases,look at a matrix and determine its value without calculations.”Object conception:“At the object level,students see the determinant as a tool for understanding and describing matrices.They understand the implications of the value of the determinant of a matrix as a way to describe a matrix.They can use the determinant of a matrix(equal to or not equal to zero)to describe properties of the elements of a matrix.”Triad development of a schema(intra,inter,trans):“A singular concept–basis.There is a basis for a space.The student can describe a basis without calculation.The student canfind different types of bases(column space,row space,null space,eigenspace)and use these values to describe matrices.”The descriptions of components of APOS along with examples illustrate that this student was able to make valid connections between the theoretical framework and the content of linear algebra.While the2094W.Martin et al./Linear Algebra and its Applications432(2010)2089–2099descriptions may not match those that would be given by scholars using APOS as a research framework, the student does demonstrate a recognition of and ability to provide examples of how understanding of linear algebra can be organized conceptually as more that a collection of facts.As would be expected,not all participants showed gains in either domain.We viewed the results of this study as a proof of concept,since there were some participants who clearly gained from the experience.We also recognized that there were problems associated with the implementation of our plan.To summarize ourfindings in relation to the research questions:•Do participants make connections between linear algebra content and learning theories?Yes,to widely varying degrees and levels of sophistication.•Do participants reflect upon their own learning in terms of studied learning theories?Yes,to the extent possible from their conception of the learning theories and understanding of linear algebra.•Do participants connect their study of linear algebra and learning theories to the mathematics content or pedagogy for their mathematics teaching?Participants describe how their experiences will shape their own teaching,but we did not visit their classes.Of the11students at one site who took the parallel courses,we identified three in our case studies (a detailed report of that study is presently under review)who demonstrated a significant ability to connect learning theories with their own learning of linear algebra.At another site,three teachers pursuing math education graduate studies were able to varying degrees to make these connections –two demonstrated strong ability to relate content to APOS and described important ways that the experience had affected their own thoughts about teaching mathematics.Participants in the workshop produced richer concept maps of linear algebra topics by the end of the weekend.Still,there were participants who showed little ability to connect material from linear algebra and APOS.A common misunderstanding of the APOS framework was that increasing levels cor-responded to increasing difficulty or complexity.For example,a student might suggest that computing the determinant of a2×2matrix was at the action level,while computation of a determinant in the 4×4case was at the object level because of the increased complexity of the computations.(Contrast this with the previously mentioned student who observed that the object conception was necessary to recognize that higher dimension determinants are computed recursively from lower dimension determinants.)We faced more significant problems than the extent to which students developed an understanding of the ideas that were presented.We found it very difficult to get students–especially undergraduates –to agree to take an additional course while studying linear algebra.Most of the participants in our pilot projects were either mathematics teachers or prospective mathematics teachers.Other students simply do not have the time in their schedules to pursue an elective seminar not directly related to their own area of interest.This problem led us to a new project in which we plan to integrate the material on learning theory–perhaps implicitly for the students–in the linear algebra course.Our focus will be on working with faculty teaching the course to ensure that they understand the theory and are able to help ensure that course activities reflect these ideas about learning.4.Continuing researchOur current Linear Algebra in New Environments(LINE)project focuses on having faculty work collaboratively to develop a series of modules that use applications to help students develop conceptual understanding of key linear algebra concepts.The project has three organizing concepts:•Promote enhanced learning of linear algebra through integrated study of mathematical content, applications,and the learning process.•Increase faculty understanding and application of mathematical learning theories in teaching linear algebra.•Promote and support improved instruction through co-teaching and collaboration among faculty with expertise in a variety of areas,such as education and STEM disciplines.W.Martin et al./Linear Algebra and its Applications432(2010)2089–20992095 For example,computer and video graphics involve linear transformations.Students will complete a series of activities that use manipulation of graphical images to illustrate and help them move from action and process conceptions of linear transformations to object conceptions and the development of a linear transformation schema.Some of these ideas were inspired by material in Judith Cederberg’s geometry text[14]and some software developed by David Meel,both using matrix representations of geometric linear transformations.The modules will have these characteristics:•Embed learning theory in linear algebra course for both the instructor and the students.•Use applied modules to illustrate the organization of linear algebra concepts.•Applications draw on student intuitions to aid their mental constructions and organization of knowledge.•Consciously include meta-cognition in the course.To illustrate,we sketch the outline of a possible series of activities in a module on geometric linear transformations.The faculty team–including individuals with expertise in mathematics,education, and computer science–will develop a series of modules to engage students in activities that include reflection and meta cognition about their learning of linear algebra.(The Appendix contains a more detailed description of a module that includes these activities.)Task1:Use Photoshop or GIMP to manipulate images(rotate,scale,flip,shear tools).Describe and reflect on processes.This activity uses an ACTION conception of transformation.Task2:Devise rules to map one vector to another.Describe and reflect on process.This activity involves both ACTION and PROCESS conceptions.Task3:Use a matrix representation to map vectors.This requires both PROCESS and OBJECT conceptions.Task4:Compare transform of sum with sum of transforms for matrices in Task3as compared to other non-linear functions.This involves ACTION,PROCESS,and OBJECT conceptions.Task5:Compare pre-image and transformed image of rectangles in the plane–identify software tool that was used(from Task1)and how it might be represented in matrix form.This requires OBJECT and SCHEMA conceptions.Education,mathematics and computer science faculty participating in this project will work prior to the semester to gain familiarity with the APOS framework and to identify and sketch potential modules for the linear algebra course.During the semester,collaborative teams of faculty continue to develop and refine modules that reflect important concepts,interesting applications,and learning theory:Modules will present activities that help students develop important concepts rather than simply presenting important concepts for students to absorb.The researchers will study the impact of project activities on student learning:We expect that students will be able to describe their knowledge of linear algebra in a more conceptual(structured) way during and after the course.We also will study the impact of the project on faculty thinking about teaching and learning:As a result of this work,we expect that faculty will be able to describe both the important concepts of linear algebra and how those concepts are mentally developed and organized by students.Finally,we will study the impact on instructional practice:Participating faculty should continue to use instructional practices that focus both on important content and how students develop their understanding of that content.5.SummaryOur preliminary study demonstrated that prospective and practicing mathematics teachers were able to make connections between their concurrent study of linear algebra and of learning theories relating to mathematics education,specifically the APOS theoretical framework.In cases where the participants developed understanding in both domains,it was apparent that this connected learning strengthened understanding in both areas.Unfortunately,we were unable to encourage undergraduate students to consider studying both linear algebra and learning theory in separate,parallel courses. Consequently,we developed a new strategy that embeds the learning theory in the linear algebra。