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# Class Note for MATH 322 with Professor Glickenstein at UA 2

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Chapters 7 8 Linear Algebra Sections 75 78 amp 81 Chapters 778 Linear Algebra Linear s stems 39f e uati s VI 0 r l Definitions V Soluiions 1 Linear systems of equations 0 A linear system of equations of the form 311X1 312X2 39 39 39 31an 131 a21X1 a22X2 32an 132 ale1 am2X2 aman bm can be written in matrix form as AX B where 811 812 39 39 39 aln X1 131 821 822 a2n X2 b2 A X B 2 3ml am2 amn Xn bm Chapters 778 Linear Algebra Linear systems of equations lnve Eigenvalues and elgem Definitions Solutions Solutions of a linear system of equations 0 Given a matrix A and a vector B a solution of the system AX B is a vector X which satisfies the equation AX B 9 If B is not in the column space of A then the system AX B has no solution One says that the system is not consistent In the statements below we assume that the system AX B is consistent 9 If the null space of A is non trivial then the system AX B has more than one solution 0 The system AX B has a unique solution provided dimNA 0 9 Since by the rank theorem rankA l dimNA n recall that n is the number of columns of A the system AX B has a unique solution if and only if rankA n Chapters 778 Linear Algebra Row operations 0 There are three types of row operations 0 Multiply a nonzero constant times an entire row r gt an 9 Exchange rows r gt rj and rj gt r 9 Add a multiple of one row to another r gt arj r 0 Row operations do not change the span of the row space 0 There are corresponding column operations which do not change the column space Chapters 778 Linear Algebra Linear systems of equ lnv i Eigenvalues a De nitions Solutions Row operations to solve linear systems Row operations can be used to solve a linear system AX B X 4y l z 10 X l y 22 2X y 52 0 Write an augmented matrix AIB 1 4 1 10 1 1 2 2 2 1 5 16 0 Use row operations to get zeroes in the first column 1 4 1 10 0 3 1 I 12 r1 l r2 0 9 3 36 2r1 r3 Chapters 778 Linear Algebra Linear systems of equations Definitions Solutions 0 Do the same with the next column 1 4 1 I 10 0 3 1 I 12 0 0 0 I 0 3r2 l r3 0 This is equivalent to the simplified system X 4y l z 10 3y z 12 0 0 0 To solve the system use back substitution Chapters 778 Linear Algebra Linear systems of equations Inverse of a matrix Eigenvalues and eigenvertors Definitions Solutions Row operations to compute the rank of a matrix 0 Given a matrix A row operations do not change the row space 0 Since the matrix 1 4 1 10 A 1 1 2 2 2 1 5 16 can be made into the matrix 1 4 1 10 A 0 3 1 12 0 0 0 0 by doing row operations the two matrices have the same row spaces o It is easy to see that the first two rows are linearly independent so the rank is 2 Chapters 778 Linear Algebra Consistency 0 The system AX B is consistent ie has a solution if equivalently 0 Gaussian elimination on the augmented matrix AIB yields a matrix of the form 31 gtllt gtllt I b1 0 32 gtllt gtllt I b2 0 0 0 33 I 0 0 0 0 0 0 0 0 0 0 ar gtilt I br 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 I 0 ie any rows reduced to all zeroes before the line are also zero after the line 9 The rank of AIB is equal to the rank of A Chapters 778 Linear Algebra Linear systems of equa ns we Definitions I z 7 Solutions tigenvalues a Inconsistency o The system AX B is inconsistent ie has NO SOLUTION if equivalently 9 Gaussian elimination on the augmented matrix AByields a matrix of the form 31 gtllt I 31 0 32 gtllt gtllt b2 0 0 0 33 gtxlt 0 0 0 0 gtllt gtllt gtllt 0 0 0 0 0 0 ar gtxlt br 0 0 0 0 0 0 0 0 br1 0 0 0 0 0 0 0 0 0 where br1 75 0 ie there is a row of zeroes before the line with a nonzero element after the line 9 The rank of AB is greater than the rank of A 9 The vector B is not in the column space of A Chapters 778 Linear Algebra Unique solutions 0 The system AX B has one unique solution if equivalently 0 Gaussian elimination on the augmented matrix AB yields a matrix of the form 31 b1 0 32 l 192 0 0 33 0 0 0 0 0 0 0 an b 0 0 0 0 0 0 0 0 0 0 0 0 ie there are all nonzero numbers on the quotdiagonalquot 9 The rank of A is equal n which is equal to the rank of which is the maximum rank so it is essential that n 2 m This means that dimNA 0 ie the nullspace is trivial 9 The columns of A form a basis for the column space Chapters 778 Linear Algebra Linear systems of equations Inverse of a matrix Eigenvalues and eigenvez tors Definitions Solutions Infinitely many solutions o The system AX B has lots of solutions if equivalently 0 Gaussian elimination on the augmented matrix AB yields a matrix of the form 31 gtllt gtllt b1 0 32 gtllt gtllt b2 0 0 0 33 0 0 0 0 gtllt gtllt 0 0 0 0 0 ar br 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 I 0 ie there are zeroes on the diagonal andor the last diagonal nonzero element is not next to the line a The rank of A is less than n This is equivalent to dimNA gt 0 9 The columns of A do not form a basis of the column space Chapters 778 Linear Algebra 0 A linear system of the form AX 0 is said to be homogeneous 9 Solutions of AX 0 are vectors in the null space of A 9 If we know one solution X0 to AX B then all solutions to AX B are of the form XX0Xh where Xh is a solution to the associated homogeneous equation AX 0 0 In other words the general solution to the linear system AX B if it exists can be written as the sum of a particular solution X0 to this system plus the general solution of the associated homogeneous system Chapters 778 Linear Algebra Definitions Determinant 017 a matrix Properties of the inverse Linear systems of n equations with n unlLnuwns Linear systems of rquations Inverse Eigenvalues and elgenvertors 2 Inverse of a matrix 0 If A is a square n X n matrix its inverse if it exists is the matrix denoted by A71 such that AA 1 A 1 A l where In is the n X n identity matrix 0 A square matrix A is said to be singular if its inverse does not exist Similarly we say that A is non singular or invertible if A has an inverse o The inverse of a square matrix A aij is given by 71 1 miq where detA is the determinant of A and C0 is the matrix of cofactors of A Chapters 778 Linear Algebra nt of a matrix ot the in 39se ems ll 17 equations with n unknowns Determinant of a matrix 0 The determinant of a square n X n matrix A aij is the scalar detA ZaraCU ZaraCU 1 j1 where the cofactor C0 is given by Cquot 1lj Ma and the minor M0 is the determinant of the matrix obtained from A by deleting the i th row and j th column of A 0 Example Calculate the determinant of A lJgti OOU39IM 0100 Chapters 778 Linear Algebra Deilnitwns Determinant of a matrix F ror 39 of the inverse Ei envalues and eigenvez tors r g U Linear ystelns m n equatlons With 11 unknuwns Properties of determinants o If a determinant has a row or a column entirely made of zeros then the determinant is equal to zero 0 The value of a determinant does not change if one replaces one row resp column by itself plus a linear combination of other rows resp columns 0 If one interchanges 2 columns in a determinant then the value of the determinant is multiplied by 1 o If one multiplies a row or a column by a constant C then the determinant is multiplied by C 0 If A is a square matrix then A and AT have the same determinant Chapters 778 Linear Algebra Properties of the inverse 0 Since the inverse of a square matrix A is given by 71 1 mlcile we see that A is invertible if and only if detA 75 0 a a o If A IS an Invertlble 2 X 2 matrix 11 12 then 821 822 A71 2 1 822 a12 detA a21 811 39 and detA 311322 321312 0 If A and B are invertible then AB 1 B lA l and A1 1 A Chapters 778 Linear Algebra Deilnitlons D 39 mt of a matrix Properties of the inverse Ei envalues and eigenvertors g J Linear systems of 7 equations With 7 unknowns Linear systems of n equations with n unknowns 0 Consider the following linear system of n equations with n unknowns 811X1 812X2 81an 131 a21X1 a22X2 82an 132 anlxl l an2X2 l l 3nan bn 0 This system can be also be written in matrix form as AX B where A is a square matrix 0 If detA 75 0 then the above system has a unique solution X given by X A lB Chapters 778 Linear Algebra inant of a rnatrir 39 ot the i else Linear systems of n equations with n unknowns Linear systems of equations summary Consider the linear system AX B where A is an m X n matrix 0 The system may not be consistent in which case it has no solution 0 To decide whether the system is consistent check that B is in the column space of A o If the system is consistent then 0 Either rankA n which also means that dimNA 0 and the system has a unique solution 3 Or rankA lt n which also means that NA is non trivial and the system has an infinite number of solutions Chapters 778 Linear Algebra Definitions Determinant of a matrix Properties of the inverse Linear systems of n equations with n unknowns Linear systems of equations Inverse of a matrix Eigenvalues and eigenvertors Linear systems of equations summary continued Consider the linear system AX B where A is an m X n matrix 0 If m n and the system is consistent then 0 Either detA 75 0 in which case rankA n dimNA 0 and the system has a unique solution 0 Or detA 0 in which case dimNA gt 0 rankA lt n and the system has an infinite number of solutions 0 Note that when m n having detA 0 means that the columns of A are linearly dependent 0 It also means that NA is non trivial and that rankA lt n Chapters 778 Linear Algebra Linear svste lnv Y Eigenvalues and elgenvectors Eigenvalues Eigenvectors 3 Eigenvalues and eigenvectors 0 Let A be a square n X n matrix We say that X is an eigenvector of A with eigenvalue A if X 75 0 and AX AX o The above equation can be re written as A AnX 0 0 Since X 75 0 this implies that A Al is not invertible ie that detA AIquot 0 o The eigenvalues of A are therefore found by solving the characteristic equation detA AI 0 Chapters 778 Linear Algebra Eigenvalues Eigenvecturs 0 The characteristic polynomial detA Al is a polynomial of degree n in A It has n complex roots which are not necessarily distinct from one another 0 If A is a root of order k of the characteristic polynomial detA An we say that A is an eigenvalue of A of algebraic multiplicity k 0 If A has real entries then its characteristic polynomial has real coefficients As a consequence if A is an eigenvalue of A so is A o It A is a 2 X 2 matrix then its characteristic polynomial is of the form A2 ATrA l detA where the trace of A TrA is the sum of the diagonal entries of A Chapters 778 Linear Algebra Linear svste Inv Eigenvalues and eIgen Eigenvalues Eigenvectors Eigenvalues continued Chapters 778 Linear Algebra Eigenvalues Eigenvectors Eigenvectors 0 Once an eigenvalue A of A has been found one can find an associated eigenvector by solving the linear system A AnX 0 0 Since NA Al is not trivial there is an infinite number of solutions to the above equation In particular if X is an eigenvector of A with eigenvalue A so is DcX where DC 6 R or C and Dc 750 0 The set of eigenvectors of A with eigenvalue A together with the zero vector form a subspace of Rquot or C EA called the eigenspace of A corresponding to the eigenvalue A 0 The dimension of E is called the geometric multiplicity of A Chapters 778 Linear Algebra Linear svste Inv Eigenvalues and elgen Eigenvalues Eigenvectors Eigenvectors continued 0 Examples Find the eigenvectors of the following matrices Each time give the algebraic and geometric multiplicities of the corresponding eigenvalues 10 oA O 5 6 17 4 1 1 o D 1 4 1 1 1 2 Chapters 778 Linear Algebra

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