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# Elementary Linear Algebra MATH 2331

GSU

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This 14 page Class Notes was uploaded by Miss Emerald Langosh on Monday October 12, 2015. The Class Notes belongs to MATH 2331 at Georgia Southern University taught by Francois Ziegler in Fall. Since its upload, it has received 16 views. For similar materials see /class/222039/math-2331-georgia-southern-university in Mathematics (M) at Georgia Southern University.

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Date Created: 10/12/15

Math 2331 Review 3 F Ziegler Name Spring 2009 Show your work to receive credit7 and your nal answer in every computation 1 Find the inverse A 1 of A 0 1 1 1 0 1 by bringing an augmented matrix into reduced echelon form 1 1 0 2 Compute the determinants A 7 1 2 0 a 3 A 7 1 0 4 5 A 7 1 654321 054321 004321 b000321 000021 000001 1234 5678 08765 4321 3 As seen sometime during the semester7 the traf c ow problem 1 400 600 2 4 300 100 5 leads to the system 1 i1 0 0 0 1 400 1 0 1 i1 0 2 7 600 CE 0 1 1 0 1 3 300 0 0 0 1 1 4 100 5 which admits the particular solution 17 75 700730070710070 a Find the rank and nullity of the the coef cient matrix A of this equation 7 by bringing A to reduced echelon form b Find a basis of the null space of A then deduce the general solution of c Find a basis of the column space S of A is the right hand side of in S7 4 Use row reduction of an appropriate matrix to nd a basis of the subspace of R3 Spannedbyltgt7lt2gt7lt gt7lt1gt 2 3 71 5 a Find the adjoint AdjA of the matrix A 1 1 2 4 5 6 b Use a to nd the determinant detA and the inverse A l 6 Let AD be the matrices representing a linear transformation T V a V in two different ordered bases B7 0 with transition matrix Q B lC a Complete the labeling of all arrows in the diagram R B R Q V V Q o R R if column z represents vector 1 in basis B7 do we have 1 B lz or z B lv if z and y represent the same vector in B and 0 do we have x Qy or y Qz is T equal to BAB l7 to C lDC7 to both7 or to neither d is AQ equal to QD7 to DQ7 to both7 or to neither if B consists of eigenvectors of T7 which matrix is diagonal A f A116 l l A 7 52 7 s 171A6 7 The matrix A i 6 i has detI 7 A l a Find the null space of 5 7 A b Find the null space of 8 7 A c Deduce7 if possible7 an ordered basis consisting of eigenvectors of A d What matrix represents A relative to your basis in c 8 The goal of this problem is to nd a square root7 of the matrix A a Compute and factor the characteristic polynomial detI 7 A b Find the eigenvalues of A c For each eigenvalue7 nd a nonzero eigenvector d Deduce an ordered basis B7 and a diagonal matrix D7 such that AB BD e Find the inverse B lg then check explicitly that A BDB l f Now nd a matrix X such that X2 A Hint First nd Y such that Y2 D then use e 9 Let V be the space of all polynomials functions of z E R and let T V a V be the linear transformation de ned by Tf 2m 2 71f derivative The eigenvectors of T are known as Legendre polynomials7 and its eigenvalues as squared angular momentum7 in quantum mechanics Which of the following polynomi als is an eigenvector of T and if so what is the corresponding eigenvalue 3 1 c 5x3 7 3x 10 The linear transformation T above has the symmetry property ltTfggt ltfTggt relative to the inner product ltfggt fill fg dx on the space of all polynomials Using the Calculus formula 1 n 0 if n is odd x dx 2 1 m if n is even compute Math 2331 Review 1 Name F Ziegler Spring 2009 Show your work to receive credlit7 and your nal answer in every computation 1 Which of the following equations are linear a 36m 7 46y 1 d zcos 300 1 7 s sin 300 y 0 639676411 1 e 96y967y 1 c 63m 7 64y 1 2 What are the sizes of the following matrices 0 s 2 c lt 104 3 2 10 b 3 i2 71 d 7 5 16 3 3 a 0 1 1 71 C gt4gt 4 Form the coef cient matrix and the augmented matrix for the system 1 7 2 7 3 7 34 5 1 1 7 2 7 Sig 7 4 75 2 721 Sig 7 43 94 75 1 5 a Bring the given augmented matrix into reduced row echelon form by performing the suggested row operations 3 0 12 i6 531 9 0 735 2 18 1 70 8 l 1 R3 R3 2R2 R2 1 1 R1 7 4R3 R2 3R1 b Write out the system of linear equations corresponding to the initial augmented matrix in a What isare its solutions7 c Write the coef cient matrix M of the system b7 and the ve elementary matrices E17 7 E5 corresponding to the ve row operations in a d Find the inverses Efl Egl of the elementary matrices in e Deduce the expression of M as a product of elementary matrices 6 a Solve the following linear system by writing its augmented matrix and using Gauss Jordan elimination reduced row echelon form x 7 22 3 z 7 3y z 76 y 7 z 3 b Write the coef cient matrix of this linear system ls it invertible Hint Use your answer to a l 72 71 l 72 71 l 0 0 7 a Find the inverse M 1 of M 3 75 72 by reducing 3 75 72 0 1 0 2 75 72 2 75 72 0 0 l b Write out and Check the matrix multiplication that veri es your answer to a Math 2331 Review 1 Name F Ziegler Spring 2009 Show your work to receive credit and your nal answer in every computation 1 Which of the following equations are linear a 36at 46y 1 d accos 30 1 5r sin 30 2 y b e3e4 1 e r 21 V c e3m e4yy21 y y b 9 2 What are the sizes of the following matrices 3 Perform the following matrix multiplications when possible 12 3 7 8 83 8o6 28 9 504 5 6 za45e 320l2 quot 4 1 49 1 2 1 4 hike W 0 3 13 WW 2 2 Mir mm we was LA 4 Form the coef cient matrix and the augmented matrix for the system l 43 I I f 5 4 2 z 5 4 c1 1 l 5 a Bring the given augmented matrix into reduced row echelon form by performing the suggested row operations 3 0 12 6 l 0 4 Z gi il 9 0 35 2 a o r I6 181 70 8 o 0 I2 1 l 3 0 12 6 t o 9 392 0 35 2 O l 0 12 33 o I 0 I2 R32R2 o o I 46 R2 1 3 O 12 6 O a o l 46 Ewen f o I 0 l7 0 b rite out the system of linear equations corresponding tr matrix in W hat isare itsisolutioms 1 3x 12 6 igtlt 3935E 2 M Ss uh w Wxw2 amp 2 3 c Write the coef cient matrix M of the system b and the ve elementary matrices Eh EF corresnmiding to the ve row operations in a L IOO V300 loo Io r Bio 10 cal 010 ool do 0390 col mum pwn iAv J lt u39ujrmr n srs s 1 Uh Aquot d Find the linverses Ef p 55 1 of theneLerneiitiar 3l39 matrices in cl E7 1 ES 53quot 54 E53 l loo loo 300 00 1 34 010 395o 430 0039 00 02 oal Ool Olo 60 Re A v e Deduce the expression of M as a product of elementary matrices nu 7 quotquot I unwulwym rvm w Hr r m r n 1 u 100100 300 loo 04 M 0390quot l0gt01000 Oo 07 00 6c 9 0 ool 39EI39EE E E E 1 Mn 559451537 checks EMW 6 a Solve the following linear system by writing its augmented matrix and using Gauss Jordan elimination reduced row echelon forrn I o z 3 1 3 I 6 91 37 223 1393 1 4gt 0 3 3R3 i Z 6 y Z Z 3 O I l 5 l 0 392 3 2 1 3 1 6 3l 6 l 0 3 2mm gt39 3 l 3 50 3 quot gt A151 3 Hawaii 0 3 quot3 1 Rm 0 0 0 2339 1910 55 rm Intu39 a gira W VOL3t hpwamlw WM Z m1lMw 7 l7 5V6WL M H s um b Write the coef cient matrix of this linear system Is it invertible Hintz Use your answer to a l O 2 r5 wot ruth39rg a lm W Jgnlwa max I 393 I a 7 mm X Man 8 am H M jM39 l ltV gtzcualux l l 100 010 001 1 2 1 l 2 1 7 a Find the inverse Jll 1 of M 3 5 2 by reducing 3 5 2 2 5 2 392 5 2 l 2 39l 390 O l l 0 0 gt 6 I i 3 I a rm gt o t2 I 393 l O 0 O 2 O l laz39ZKg O 0 I 39539 l l BYTEI 39 l o 3941 I Kaiij I O O Elf 5 R 9 lt6 a 2 a gt 1 HZRL l a a l u 12 O 39l b Write out and check the matrix multi lication that veri es your answer to a 3 we 1 2 cm lo Ha 31 gigit z 0 l r 9 1 My 2 5 3992 5 l lnwlo 22 2S2 r Math 2331 Review 2 Name F Ziegler Spring 2009 Show your work to receive credit7 and your nal answer in every computation 1 Find the determinants Support your answer with an explicit calculationargument 2341 a112 456 12414 24728 b 01411 56 78 3251 014 2 C 00271 000 3 1234 01610 d 0a10 4321 2 Use a determinant to decide for which values of a the system ax1iay 1 has a unique solution 17aay 3 3 a Find the adjoint AdjA of the matrix A WOH b Use a to nd the determinant detA C Use a and b to nd the inverse A l 1 4 Which of the following sets of vectors x lt f are subspaces of R7 Justify 7 a all x such that 1 2 0 b all x such that 1 3x2 3 C all x such that an 0 5 Which of the following vectors7 if any7 is in the null space of A 1011 g 1 A2113u1v1w 1022 71 0 2 1 3 1 1 0 1 0 6 The matrix A 1 71 0 1 row reduces to B 0 1 1 0 1 1 2 1 0 0 0 1 a What is the rank of A b What is the nullity of A C Find a basis of the row space of A 1 Find a basis of the column space of A e Find a basis of the null space of A f Use 1 to decide if the equation Aw 2 has a solution7 without solving it 7 a Use row reduction to compute the rank and nullity of A wwyb OHH HHM HOH

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