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# LINEAR ALGEBRA MATH 544

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This 4 page Study Guide was uploaded by Cassidy Grimes on Monday October 26, 2015. The Study Guide belongs to MATH 544 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/229553/math-544-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

Math 544 Exam 3 Information Exam 3 will be based on 0 Sections 41 45 47 and 52 53 o The corresponding assigned homework problems see httpwwwmathsceduNboylanSCCoursesmath5443544htm1 At minimum you need to understand how to do the homework problems 0 Lecture notes 1029 1126 0 Quizzes 7 10 Topic List not necessarily comprehensive You will need to know how to de ne vocabulary wordsphrases de ned in class 41 The eigenvalue problem for 2 X 2 matrices De nition and computation of eigenvalues and eigenvectors for 2 X 2 matrices 42 Determinants and the eigenvalue problem De nition and computation of determinants of matrices A E MatanOR Computation of determinants by expansion across rows or down columns using minors and cofactors What is the minor and cofactor associated to a matrix entry 17 of A Properties of determinants for example 0 detAB detAdetB o A E MatanOR is singular ltgt detA 0 7 i 1 0 HA is invertible then detA 1 i W What is the determinant of a triangular matrix 43 Elementary operations and determinants Important properties detA detAT detcA c detA for c a 0 in R Effects of elementary row and column operations on the computation of a determinant o Interchanging two rows or two columns changes the sign of the determinant o If the row operation 1 f3 gt gt gRi k 74 0 transforms matrix A into matrix B then detA kdetB In effect you are factor ing k out of the 2th row of A Similarly if the column operation 1 CW 1mm transforms matrix A into matrix B then detA kdetB o A row operation of the form EHR kRj7 kill 1 does nothing to the determinant Similarly a column operation of the form 01702 ka ka O7 ia j does nothing to the determinant 44 Eigenvalues and the characteristic polynomial The de nition and computation of the eigenvalues of a matrix A E Matnm R ie computation of the characteristic polynomial pt detA7tIn the algebraic multiplicity of an eigenvalue A is the number of times the factor If 7 A occurs in the characteristic polynomial pt o If A is an eigenvalue of A and k 2 0 is an integer then Ak is an eigenvalue of A 0 HA is an eigenvalue ofA and 04 E R then A 04 is an eigenvalue ofA 04 o If A is invertible and A is an eigenvalue of A then i is an eigenvalue of A l o If A is an eigenvalue of A then it is also an eigenvalue of AT 0 A matrix A has 0 as one of its eigenvalues if and only if it is singular What are the eigenvalues of a triangular matrix 45 Eigenspaces and eigenvectors The de nition and computation of the eigenvectors ofa matrix A E Matnm R If A is an eigenvalue ofA E Matnm R then the eigenspace associated to A is EA NullA 7 AI and the geometric multiplicity of A is the dimen sion of EA ie the nullity of A 7 A1 The relationship between algebraic and geometric multiplicities is l S geometric multA S algebraic multA De nition of a defective matrix a matrix A is defective if A has at least one eigenvalue whose geometric mult is strictly less than its algebraic mult ie there is an eigenvalue A with geom multA lt alg multA Important fact Eigenvectors associated to distinct eigenvalues are linearly independent As a consequence if A E Matnm R is not defective then A has n linearly independent eigenvectors and these eigenvectors form a basis for R In particular if A has n distinct eigenvalues then A is not defective 47 Similarity transformations and diagonalization Matrices A and B E MatanOR are similar if there is an invertible matrix S for which B S lAS A matrix A is diagonalizable if it is similar to a diagonal matrix D If A and B are similar they have the same 0 characteristic polynomial p At p3 However the converse is not true If pAt pg 75 then it is not always true that A and B are similar 0 eigenvalues and algebraic multiplicities but the corresponding eigenvectors are typ ically different If B S 1AS so A and B are similar and if x is an eigenvector of B associated to A so Bx Ax then S is an eigenvector ofA associated to A so ASx Criterion for diagonalizability The diagonalizability of A is equivalent to o A has n linearly independent eigenvectors the maximum possible 0 A is not defective ie the geometric and algebraic multiplicities agree for all eigen values of A So a matrix A is either defective or diagonalizable If A is diagonalizable then there is an invertible matrix S and a diagonal matrix D for which D S lAS How do you nd the matrices S and D 0 Compute the eigenvalues of A and their algebraic multiplicities Suppose that the dis tinct eigenvalues ofA are A1 7 M Compute bases 81 781 for the eigenspaces EA 7 EM The dimension of EA is the geometric multiplicity of A If for all 239 alg mult geom mult 7 then A is diagonalizable o If A is diagonalizable form the set 8 vi17 wn consisting of all the basis vectors for the eigenspaces of A Then the invertible matrix S which diagonalizes A is SVs71lv72l W7 So we have D S lAS where D is a diagonal matrix with diagonal entry A and A is the eigenvalue ofA associated to the eigenvector w Aw Aiw If A is diagonalizable and k 2 0 is an integer how can you compute Ak Here s how A diagonalizable implies that for some invertible matrix S D S 1AS is diagonal We then have Dk S lASk S leS Moving the S s to the left side we obtain SDkS 1L Ak So ifyou know S and S 1 it is easy to compute Dk ifD is diagonal you can compute Ak Orthogonal matrices Their de nition and basic properties 0 Q E MatanOR is orthogonal if and only if its rows and columns form orthonormal bases for R o If you rearrange the rows or columns of an orthogonal matrix the resulting matrix is still orthogonal o If Q is orthogonal then V E R Multiplication by Q preserves length Vi 7 E R Q Q57 3 Multiplication by Q preserves the angle between vectors detQ i1 0 A E MatanOR is symmetric if and only if it is orthogonally diagonalizable ie 3Q orthogonal such that Q lAQ D is diagonal 52 Vector spaces The de nition of vector space a set V and a scalar eld F together with an addition operation on V and a scalar multiplication operation in particular the ten vector space axioms 2 closure axioms 4 axioms for vector addition 4 axioms for scalar multiplication Examples of vector spaces MathnOR Pn Check whether a set V together with an addition and scalar multiplication is or is not a vector space 53 Subspaces Determination of whether or not certain subsets of MathnOR or Pn are subspaces

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