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# Linear Algebra MA 26500

Purdue

GPA 3.97

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This 20 page Study Guide was uploaded by Dorothea Bode on Saturday September 19, 2015. The Study Guide belongs to MA 26500 at Purdue University taught by Staff in Fall. Since its upload, it has received 46 views. For similar materials see /class/208119/ma-26500-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

STUDY GUIDE FOR MA 265 LINEAR ALGEBRA This study guide describes brie y the course materials to be covered in MA 265 In order to be quali ed for the credit one is expected not only to know these materials but also to demonstrate the skills to solve the quantitative nd numerical problems The current textbook used in MA 265 is as of April 16 2001 0 Introduction to Linear Algebra with Applications 6th Edinition by Bernard Kolman where most of the materials discussed below can be found and studied 1 Linear Systems 0 De nition of a linear system 0 Method of elimination Elementary Operations Description of the set of solutions One should know how to solve a linear system of equations by the method of elim ination GaussJordan elimination method using elementary operations which leads to the complete descripion of the set of solutions 2 Matrices 0 De nition of a matrix 0 Matrix Operations Addition Scalar multiplication Transpose Multiplication of matrices 0 Properties of Matrix Operations 0 Inverse Matrix How to nd the inverse via GaussJordan method The formula A 1 d 1 adjA etA One should know the basic opeations of matrices addition scalar multiplica tion transpose and matrix multiplication We observe the analogy with the usual operations of numbers while the matrix multiplication is not commutative ie AB BA in general The inverse A 1 of a matrix A analogous to the notion of the reciprocal 10 of a number a is studied One should know how to com pute the inverse of a matrix A when it exists One is to apply the GaussJordan Typeset by AMSTEX 2 STUDY GUIDE reduction to the augmented matrix AI a 1 A I Another is to apply the for mu a A 1 delz adjA7 Which can be obtained as an easy application of the basic proerties of the determinant of a matrix 3 Solutions of Linear Systems in terms of Matrices o Augmented Matrix 0 Reduced Row Echelon Form RREF o GaussJordan reduction Description of the set of solutions Via RREF o Homogeneous System Dimension of the set of solutions the null space 0 NonHomogeneous System Reduction to the associated homogeneous system Via a particular solution One should know how the problem of solVing a linear system of equations can be translated into the problem of transforming a given augmented matrix into Reduced Row Echelon Form Via the GaussJordan reduction using elementary roW operations One should be able to read off7 by looking at RREF such information as the dimension of the null space the space of the set of solutions for a homogeneous system and be able to describe the Whole set of solutions for a nonhomogeneous system giVen a particular solution and the description of the null space 3 Determinants 0 De nition of a determinant Permutation The number of inVersion 0 Properties of determinants o Cofactor Expansion A adjA detA I Cramer s rule One should know how to compute the determinant of a matrix Starting With the Very de nition using the language of permutaion odd and eVen permutations and the number of inVersions etc7 Via the study of its basic properties7 one is led to the cofactor expansion Cofactor expansions can be collectiVely described as the formula A adjA det A I Which then proVides the formula for the inVerse of a matrix A An important application is Cramer s rule 4 Linear Af ne Geometry of R o Vectors in R 0 Operations among Vectors ion Scalar multiplication Dot Product Orthogonality in terms of dot product 0 Lines and planes in R2 and R3 Parametric and symmetric equations of a line Equation for a plane With a giVen normal Vector and a point on it MA 265 3 One should be able to describe the linear af ne geometry of R in terms of vectors Aside from the basic operations of addintion and scalar multiplication the dot product is the most important one to study providing eg the orthogonality criterion of two vectors The emphasis is put on how to nd the parametric and symmetric equations of a line in R2 and R3 as well as how to nd the equation of a plane in R3 satisfying certain conditions 5 Vector spaces 0 De nition of an abstract vector space 0 Subspaces 0 Linear Independence and Dependence 0 Basis and Dimension Column and row space of a matrix Rank of a matrix 0 Orthonormal basis GramSchmidt process Orthogonal complements Applications to the method of least squares This material is the most abstract and hardest of all though once understood it will be the powerfull ally of yours in analysing a wide varieties of mathematical and physical phenomena ranging from differential equations to optimization One should know the de nition of a vector space over R and subspaces abstracting the notion of R he notion of linear independence and dependence clari es eg what we mean by idependent or free varibles in describing the set of solutions for a linear system One should be able to nd a basis a set of linearly independent vectors which span for a given subspace eg the column space or row space of a matrix Through the GramSchmidt process one should be able to compute and obtain an orthonormal basis out of a given basis One should be able to compute the projection of a vector onto a subspace and onto its orthogonal complement As an important application we discuss the least square t solutions notably how to draw the least square t line for given data points 6 Linear Transformations 0 De nition of a linear transformation 0 Kernel and range of a linear transformation 0 Matrix representing a linear transformation 0 Coordinates and change of basis A linear transformation between R and R is simply a map de ned by multipli cation of an m X 71 matrix called the standard matrix of the linear transformation One should know how to nd the standard matrix for rotation re ection or com bination of these A linear combination between vector spaces is similarly de ned once we identify them with R an m by choosing bases The choice of bases corresponds to certain transformation of the standard matrix 7 Eigenvalues and Eigenvectors 0 De nition of eigenvalues and eigenvectors Characteristic polynomial 4 STUDY GUIDE 0 Diagonalization iagonalization of a matrix similar to a diagonal matrix Diagonalization of a symmetric matrix Eigenvalues and eigenvectors v of a square matrix A is characterized as the solutions to the equation Av v with v 0 On a more concrete level one should know how to compute the eigenvalues as roots of the characteristic polynomial detAI 7 A and compute then the eigenvectors as solutions to AI 7 Av 0 One of the main applications one should know is the diagonalization problem Given a square matrix A nd an invertible matrix P such that P lAP D is a diagonal matrix Though this is not always possible when A has distinct eigenvalues or when A is symmetric we can give a speci c algorithm to nd P in the case of a symmetric A we may require further for P to be orthogonal 8 Linear Differential Equations 0 Solutions to linear differential equations with diagonalizable matrices real eigenvalues real solutions with complex eigenvalues Complex numbers and Euler s formula 0 Solution of the form X em for A exponential of a matrix As an application of the diagonalization problem and its solution we discuss how to solve linear differential equations While the more detailed treatment should be given in MA 266 we give a complete discussion when the representation matrix is diagonalizable One should know how to write down the general REAL solution w en the representation matrix has COMPLEX eigenvalues via Euler s formula The solution in the case when the representation matrix is not diagonalizable can be found via the discussion of the exponentials of matrices 9 Matlab computer 0 Understanding of basic commands 0 Matrix operations in Matlab 0 Elementary Row Operations in Matlab o Vectors in Matlab The use of Matlab is an essential and active part of MA 265 though the real use of computers in the exams is not reuired or practiced under the current curricu lum One should know the basic commands such as rref and inv and be able to interpret the output depending on the situation The skills to read the necessary information off the output of Matlab will be tested Math 265 Linear Algebra Credit Exam Practice Student Name print Student ID Do not write below this line Please be neat and show all work Write each answer in the provided box Use the back of the sheets and the last 3 pages for extra scratch space Return this entire booklet to your instructor No books No notes No calculators Problem Max pts Earned points 1 20 2 8 3 8 4 8 5 8 6 8 7 8 8 8 9 8 10 8 11 8 Section I 100 13 10 14 10 15 10 Section II 30 16 20 17 15 18 15 19 20 Section III 70 TOTAL 200 Section I Short problems No partial credit on this part but show all your work anyway It might help you if you come close to a borderline Please be neat Write your answer in the provided box 0 3 649 10 305 1 2 131 0120 3 11t1sg1venthatA 2 3 0 3 1rrefA 0 0 0 1 0 and 145 9 7 00000 100 0 010 5 rrefAT001 2 000 0 000 0 a Find the rank of A b Find a basis for the null space of A C Find a basis for the column space of A We require that you choose the vectors for the basis from the column vectors of A d Find a basis for the row space of A We require that you choose the vectors for the basis from the row vectors of A 2 Determine the values of a so that the following linear system has no solution 261 2562 x3 a 1131 2 1113 33121 41152 02 2 L393 II II F r I 3 Find the standard matrix for the linear transformation L R3 gt R2 such that 1 0 0 L 0 t 1 ms 0 32 0 0 1 4 Determine the values of a so that the line whose parametric equations are given by x 3t yZQ t z1at is parallel to the plane 3x 5yz30 5 Find the symmetric equations of the line which is the intersection of the following two planes xy z23x4yz5 6 Compute the inverse of the matrix A Or Av l r aor a Dol A 7 E is a 3 X 3 matrix of the form 1 8 3 E a y z 3 7 2 Given detE 5 compute the determinant of the following matrix 1 y 2 F 1 8 3 34x 74y 2 142 8 Find the matrix G such that 9 Find the dimension of the subspace V spanv1 v2 v3 124 in R3 Where 1 2 0 1 U1 0 73902 1 7113 0 U4 1 1 2 1 1 7 10 Find the projection Projwv of the vector v 0 onto the subspace W spanned by 11 We have a subspace W in R4 spanned by the following three linearly independent vectors 1 2 3 1 1 3 ul 2 0 711 0 u3 0 0 0 2 Find an orthonormal basis of W 12 13 Section II Multiple choice problems For Problems 12 through 15 circle only one the correct answer for each part No partial credit Let A be a 3 x 3 matrix with detA 0 Determine if each of the following statements is true or false a Ax 0 has a nontrivial solution True False b Ax b has at least one solution for every b True False c For every 3 x 3 matrix B we have detA B detB True False d For every 3 x 3 matrix B we have detAB 0 True False e There is a vector b in R3 such that rankA b gt rankA True False For each of the following sets determine if it is a vector subspace a The set of all vectors 31 372 m3 4 in R4 with the property 2331 2 2 0 313 m4 2 0 Yes No b The set of all vectors 331 332333 in R3 with the property 51 2 01122 gt 0333 2 0 Yes No c The set of all vectors 31 2 3174 in R4 with the property at 3 23 2 1 Yes No cl The set of all vectors of the form a b 12a 3C 11 ca b c 2 in R4 where a b and c are arbitrary real numbers Yes No dx e The set of all solutlons to the llnear system of differentlal equatlons d Ax 5 4 whereA 1 1 Yes No 14 For the problems a b and c7 determine if the given set of vectors is linearly inde pendent or linearly dependent 2 39 392 a I 2 1 Independent Dependent 3 0 39 1 39 39 2 0 b 0 1 1 Independent Dependent 1 1 1 39 1 39 39 3 0 4 c 2 0 1 4 Independent Dependent 5 11 1 5 For the problems d and e determine if the given set of vectors spans R3 7T 3 d 27r 1 span not span 17r 1 1 4 7 1 e 2 5 8 0 span not span 3 6 9 0 Section III Multi Step problems LShow all work no work no credit and display computing steps Write clearly 15 Let 8 15 a Find the eigenvalues and compute an eigenvector for each eigenvalue A 15 28 b Find an invertible matrix P and a diagonal matrix D such that P 1AP D c Compute A37 16 Find the least squares t line for the points 271 1237 072 173 271 17 Let de39l dt A 1 d2 2 dt be the linear system of differential equations where 3 5 A l5 3 a Find the eigenvalues and nd an eigenvector for each eigenvalue for A Note The eigenvalues are COMPLEX valued b Find the general REAL solution to the linear system of differential equations 18 Let d 61 21131 i 511 d g 2 3731 1112 31173 dLIT3 2 dt 1 be a linear system of differential equations a Find the eigenvalues and nd an eigenvector for each eigenvalue for the coef cient matrix of the linear system of differential equations b Find the general solution to the linear system of differential equations c Find the solution to the initial value problem 2710 4120 16m30 0 a a A1 7371 KIA 265 FINAL EXAXI Spring 2001 ANSWERS 3 75 2 3 1 a 3 b 1 0 c A Ag A4 d A Ag Ag 0 0 0 1 1 2 0 l 7 3 11 1 2 175 3 0 5 i2 4 1778 5 0 77 472 0 1 0 74 6 2 7 1 7 5 8 r 2 J 9 5 i1 1 1 quot 1 1 3 g 10 0 11 TJ 5 0 O O 1 0 0 12 a True 1 False c False d True e True 13 a Yes 1 No c No d Yes e Yes 14 a Independent 1 Dependent c Dependent d not span e span 2 7 2 7 1 0 7 J J J 15 am 11 My 112 11 1 4111 0 1 16 y 2 17 a135i71 M A2 37517112 sinSt g7 c0551 b XV 0163 626 Sin 5t cos 5t 3 75 75 73121711 1 A3571 73 1 5 1 3 75 75 b xt ale 339S 73 026quot 1 0365quot 73 1 5 1 5 5

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