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# Elementary Linear Algebra MA 35100

Purdue

GPA 3.97

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This 4 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 35100 at Purdue University taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/208139/ma-35100-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

a 01 K l Ordered basis 8 X17X27gtH MA 351 Fall 2004 REVIEW SHEET EXAM 7 3 Linear transformations T V a W Matrix transformations TA R a Rm where TAX AX for some mgtltn matrix A mapping properties ofTA line segments mapped to line segments or points Matrix multiplication Compositions of transformations composition of matrix transformation represented as a product of matrices TB R4 a R and TA R a Rm then TA 0TBgtltXgt ABX RANK OF PRODUCTS THEOREM lfA is m gtlt n and B is n gtlt q then rank AB 3 rank A and rank AB 3 rank B Properties of matrix multiplication ABC ABC7 ABt B fA f7 AB C AB AC7 B CD BD CD provided all products are de ned Note In general AB 7E BA lnverse transformations invertible matrices nding A 1 using row reduction if A and B are invertible matrices then AB 1 B lA l INVERTIBILITY THEOREM Let A be an n gtlt n matrix Then A is invertible ltgt rank A n 7Xn for V if X E V and X ciXl 0ng c Xn 551 then the B coordinate vector of X is the vector X E R given by X Point and Coordinate matrices for V R PB XL7 X27 Thus X PBX and X 03X 7X4 and CB Pgl Point transformation for V wrt ordered basis B T R a V where 1 552 I x1X1x2X2wann 11 Coordinate transformations for V wrt ordered basis 3 Tg V a R where Tg1X1 Jr 2X2 Jr Jr Recall Tg T113 00 CD 11 12 13 14 15 16 17 18 MATRIX REPRESENTATION FOR LINEAR TRANSFORMATION THEOREM IfL V 7 W is a linear transformation with By X1X2 Xn and BW 517527 Ym ordered bases for V and W respectively then there is a unique m gtlt n matrix M such that LX MX where X is the By coordinate vector of X and LX is the 8W coordinate vector of LX The matrix M is given by M LX1 LX2 LXn For the special case V R and W Rm can also nd M using M 05W A Cgvl lsomorphic vector spaces isomorphisms L V 7 W 10 ISOMORPHIO VEOTOR SPACES THEOREM dim W n If V is isomorphic to W and dimV n gt Determinants of 2 gtlt 2 and n gtlt n matrices ijth cofactor of an n gtlt n matrix A ie Ci 71ij det M where Mij is the n 7 1 gtlt n 7 1 matrix obtained by deleting the 2 row and jth column of A the LaplaceCofactor expansion for the determinant TL detA Z aikCik cofactor expansion along 2 row k1 TL detA Z aimCk cofactor expansion along jth column k1 Properties of determinants RowColumn Exchange Property RowColumn Scalar Property RowColumn Addition Property More properties of determinants detAB detA detB7 detAt detA7 detcA c detA etc CRAMER7S RULE Let A A17 A2 Ak An be an n gtlt n invertible matrix The 551 unique solution X to the system AX B is given by 11 detlAi7 A27 W7A77l k 1 2 5 detA 7 7 7 7n Eigenvalue A of and n gtlt n matrix A detA 7 AI 0 eigenvector X of A corresponding to A AX AX where X E R is a non zero vector Characteristic polynomial of the matrix A pgt detA 7 AI W Let A be an n gtlt n matrix Then A is an eigenvalue of A ltgt pgt 0 Note that A satis es the matrix equation pA 0 Eigenspace ofA corresponding to A ie W X E R AX AX basis for the eigenspace An n gtlt n matrix A is diagonalizable if it has Q linearly independent eigenvectors if not A is de cient DIAGONALIZATION THEOREM Let A be an n gtlt n diagonalizable matrix and let 8 Q7L7 Q2 Qn be Q linearly independent eigenvectors where Qk corresponds to the eigenvalue M Then A QDQ 1where Q Q7L7 Q2 Qn and D diaggt1gt2 An 2 1 2 00 a 01 00 CD LetA PRACTICE PROBLEMS 1 2 3 0 If A is a 3 gtlt 3 matrix and At 1 0 1 3 solve the system AX 71 1 1 71 2 Let A and B be n gtlt n invertible matrices Prove the following a ABA l3 ABSA l b A3157 AB QA l C 43W 4 53 5 4 2 Let L R3 a R3 be the linear transformation de ned by LX AX7 where A 4 5 2 J 2 2 2 Find the matrix representation M of L with respect to this basis for R3 8 lil ljl l 31 1111011721 Use this basis for both the domain and target vector spaces Using the standard bases for 732 and 733 nd the matrix representation M for the linear transformation L 733 a 732 given by La by 0x2 data b 2095 361952 If A is a 3 gtlt 3 matrix and lAl 73 then det2A 7 If A and B are 4 gtlt 4 matrices where lAl 2 and det2B 73 then nd l7BA 1l 1 IfA t bl thtth 17 IS IHVBI 1 e prove a e detA 2 0 2 0 x1 1 l 1 1 1 1 x2 1 2 Solve for 51 and 63 for the System 1 0 3 73 x3 7 0 39 1 0 0 5 x4 1 1 1 4 1 39 a Find eigenvalues and corresponding eigenvectors of A b Find a matrix Q and a diagonal matrix D such that A QDQ l c Compute A and A s d Find a matrix equation which A satis es 3 10 BOOK PROBLEMS Page 162 3 5 1319b2326 Page 186 33 Page 197 2 abchp 12 19 20 23a 30 31 Page 233 4 b 5a 6a 13ad 16ce Page 250 1 a i 4 Page 259 1 abc1213 Page 268 1 2 Page 278 1ab 4ace 10a 1112 14

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