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# 216 Class Note for MA 51100 at Purdue

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Date Created: 02/06/15

Linear Algebra and Its Application MATH 511 Purdue University Summer 2006 n Inverse of a matrix A Definition Let A be an n x n matrix The inverse is the n x n matrix B such that AB In BA and is denoted by A 1 n Inverse of a matrix A Definition Let A be an n x n matrix The inverse is the n x n matrix B such that AB In BA and is denoted by A 1 The following are equivalent 0 A is invertible o rankA n o The column vectors of A are linearly independent 0 A linear system AX b has a unique solution for any h o The linear system AX O has only trivial solution 0 detA 7 O How to find the inverse of a matrix A 0 Find the RREF of the matrix A In Himv 1116 we mmxrgrg i f WNW 0 Find the RREF of the matrix A In How to solve a consistent linear system AX b Gaussian Elimination 9 Use REF of A i b and backsubstitution 0 Use RREF ofA i b How to solve a consistent linear system AX b Gaussian Elimination 9 Use REF of A i b and backsubstitution 0 Use RREF ofA i b Suppose that A is invertible 0 X A 1b o Cramer s Rule m LH i Least squares solution Let AX b be an m x n linear system Definition o The normal equation is ATA ATb o A solution to the normal equation is called a least squares solution ii Least squares solution Let AX b be an m x n linear system Definition o The normal equation is ATA ATb o A solution to the normal equation is called a least squares solution Suppose that the columns of A are linearly independent o NA is invertible and 2 ATAHATb o QRfactorize A and 2 Fr1 0 n Vector space A vector space V ea Q is a set V with two operations ea Q w o Foranyuve Vueave V ueavveau ueaveawueaveaw There exists 9 e V such that u ea 9 ufor any u e V For any u e V there exists 7u e V so that u ea iu O For any u e V and any scalar c c Q u e V cueavc ueac v cd uC Ud L c d ucd u 1 uu 000000000 n Examples of vector spaces o R with ordinary addition and scalar multiplication o C with ordinary addition and scalar multiplication o The set P of all polynomials of degree at most n with ordinary addition and scalar multiplication o The set Mm of all m x n matrices with ordinary addition and scalar multiplication o The set CFOWO of all continuous functions with ordinary addition and scalar multiplication oThesetVreRlrgt0withu vuvandc 1uc n Subspace Let V be a vector space A nonempty subset W of V is a subspace of V if it is closed under addition and scalar multiplication n Subspace Let V be a vector space A nonempty subset W of V is a subspace of V if it is closed under addition and scalar multiplication The subset W of M consisting of aquot symmetric matrices is a subspace of Mn Let A B be n x n symmetric matrices and c any real number Then we have A BT AT 57 A 5 cAT CAT cA m Lm Basis and dimension Definition A set 3 V1 Vd is a basis of a vector space V if 0 V17Vd span V 0 v1 Vd are linearly independent In this case the dimension of V is d n Basis and dimension Definition A set 3 V1 Vd is a basis of a vector space V if o V17Vd span V 0 v1 Vd are linearly independent In this case the dimension of V is d Let V be a vector space of dimension d 0 We can find at most d linearly independent vectors in V 0 We need at least d vectors to span V o If d vectors in V are linearly independent they form a basis for V o If d vectors in V span V they form a basis for V Four fundamental subspaces of a matrix A Let A be an m x n matrix with rankA r o The column space ofA is the subspace of Rm spanned by the columns of A and has dimension r Four fundamental subspaces of a matrix A Let A be an m x n matrix with rankA r o The column space ofA is the subspace of Rm spanned by the columns of A and has dimension r o The nullspace of A is the subspace of R consisting of solutions to AX O and has dimension n 7 r Four fundamental subspaces of a matrix A Let A be an m x n matrix with rankA r o The column space ofA is the subspace of Rm spanned by the columns of A and has dimension r o The nullspace of A is the subspace of R consisting of solutions to AX O and has dimension n 7 r o The row space of A is the subspace of R spanned by the rows of A and has dimension r Four fundamental subspaces of a matrix A Let A be an m x n matrix with rankA r o The column space ofA is the subspace of Rm spanned by the columns of A and has dimension r o The nullspace of A is the subspace of R consisting of solutions to AX O and has dimension n 7 r o The row space of A is the subspace of R spanned by the rows of A and has dimension r o The left nullspace of A is the subspace of Rm consisting of solutions to ATX O and has dimension m 7 r n Linear transformation Definition Let V and W be vector spaces A map T V a W is called a lineartransformation if Tc1v1 czvz c1Tv1 62TV2 for any v1 V2 6 V and any scalars 0102 n Linear transformation Definition Let V and W be vector spaces A map T V a W is called a lineartransformation if Tc1v1 czvz c1Tv1 62TV2 for any v1 V2 6 V and any scalars 0102 Representation of a linear transformation 0 For any m x n matrix A there is a linear transformation T Rquot HR quot defined to be Tv Av n Linear transformation Definition Let V and W be vector spaces A map T V a W is called a lineartransformation if Tc1v1 czvz c1Tv1 62TV2 for any v1 V2 6 V and any scalars 0102 Representation of a linear transformation 0 For any m x n matrix A there is a linear transformation T Rquot HR quot defined to be Tv AV 0 For any lineartransformation T V a W between finite dimensional vector spaces there exists a matrix A such that TVg Avg where 18 are bases for V W respectively How to find a representation of a linear transformation Let T P1 a P2 be the linear transformation defined to be Tpt tpt p0 Let B t 1 t7 1 and B t2 1J7 1 t 1 be bases of P1 and P2 respectively How to find a representation of a linear transformation Let T P1 a P2 be the linear transformation defined to be Tpt tpt p0 Let B t 1 t7 1 and B t2 1J7 1 t 1 be bases of P1 and P2 respectively t71 t1 oTt1t2t11t21 1 t717 t1 oTt71t27t711t2 volt Nl How to find a representation of a linear transformation Let T P1 a P2 be the linear transformation defined to be Tpt tpt p0 Let B t 1 t7 1 and B t2 1J7 1 t 1 be bases of P1 and P2 respectively oTt1t2t11t21t71t1 oTt71t27t711t21 t717t1 1 1 oTt1gz12Tt1gz 12 12 732 1 1 o The representation is 12 12 12 732 n Standard inner product The following are standard inner products 0 InR ltvWgtvT W 0 In C lt vw gt VHW 0 In Clo lt f7g gt o in Mm lt AB gt trATB Orthogonal complement WL v E V v J W Let W be a subspace of an inner product space V o W m WL O o dimW dimWi dimV o For every vector v e V there is a vector W e W and u e WL such that v W u In this case W projWv and u v 7 projWv Orthogonal complement WL v E V v J W Let W be a subspace of an inner product space V o W m WL O o dimW dimWi dimV o For every vector v e V there is a vector W e W and u e WL such that v W u In this case W projWv and u v 7 projWv Let A be a matrix 0 cAi NAT o ATL NA Projection Let W be a subspace of R with standard inner product If B W17 Wd is an orthogonal basis then the projection ofv e R onto W is W1TV W2Tv WdTv promV T W1 T W2T7Wd W1 W1 W2 W2 Wd Wd Projection Let W be a subspace of R with standard inner product If B W17 Wd is an orthogonal basis then the projection ofv e R onto W is W1TV W2Tv WdTv PTOJWV T W1 T W2 77Wd W1 W1 W2 W2 Wd Wd If B u1 ud is a basis let A be a matrix whose columns are U1 ud Then the projection of v e V onto W is projWV AATA 1ATV L1H Gram Schmidt Process Let v1 vm be linearly independent vectors in R W1 V1 W 7 v 7 WTWW 2 7 2 WW 1 WTV3 WTV3 7 1 7 2 W3 7 V3 WW 1 WJW2W2 wTv WT Vm 7 7 1 m 7 7L Wm 7 Vm WW1 ngwmiW m71 o W17W2 Wm are orthogonal 1 1 1 o 7W 7W 7W are orthonormal HW1H 1 HW2H 2 HWmH m Eigenvalues eigenspace and multiplicities Let A be an n x n matrix 0 Any number of which Av v has a nontrivial solution is called eigenvalue of A o The corresponding nonzero vector v such that Av v is called eigenvector ofA associated with Eigenvalues eigenspace and multiplicities Let A be an n x n matrix 0 Any number of which Av v has a nontrivial solution is called eigenvalue of A o The corresponding nonzero vector v such that Av v is called eigenvector ofA associated with o The eigenspace of A associated with is the nullspace of A 7 I Eigenvalues eigenspace and multiplicities Let A be an n x n matrix 0 Any number of which Av v has a nontrivial solution is called eigenvalue of A o The corresponding nonzero vector v such that Av v is called eigenvector ofA associated with o The eigenspace of A associated with is the nullspace of A 7 I o The characteristic equation ofA is detA 7 I 7 71quotA 7 mm 7 A239quot2 7 MW where m is the algebraic multiplicity of Eigenvalues eigenspace and multiplicities Let A be an n x n matrix 0 Any number of which Av v has a nontrivial solution is called eigenvalue of A o The corresponding nonzero vector v such that Av v is called eigenvector ofA associated with o The eigenspace of A associated with is the nullspace of A 7 I o The characteristic equation ofA is detA 7 I 7 71quotA 7 mm 7 A239quot2 7 MW where m is the algebraic multiplicity of o The geometric multiplicity of is the dimension of the eigenspace associated with n Diagonalizable matrix Let A be an n n matrix The following are equivalent 0 A is diagonalizable o A has n linearly independent eigenvectors o For each eigenvalue ofA the algebraic multiplicity of is equal to the geometric multiplicity of n Diagonalizable matrix Let A be an n n matrix The following are equivalent a A is diagonalizable o A has n linearly independent eigenvectors o For each eigenvalue ofA the algebraic multiplicity of is equal to the geometric multiplicity of 1 2 LetA 1 0 J with detA7I3 71371273 1 3 1 0 1 1 0 0 O 72 0 such that S 1AS O 1 O 0 1 1 0 0 3 Application of diagonalizable matrix Suppose that A is diagonalizable say S 1AS Q o Ak SQKS 1 Application of diagonalizable matrix Suppose that A is diagonalizable say S 1AS Q o Ak SQKS 1 o The general solution to a homogeneous differential equation system X t AXt is Xt eA XO SemS 1XO n Symmetric matrix Let A AT be a symmetric matrix 0 All the eigenvalues are real numbers 0 Eigenvectors belonging to distinct eigenvalues are orthogonal o It is orthogonally diagonalizable There exists an orthogonal matrix Q and a diagonal matrix Q such that Q lAQ Q QTAQ Hermitian unitary normal matrices Definition Let A be an n x n complex matrix 0 A is called Hermitian if A A o A is called unitary if AHA I o A is called normal if AHA AAH Hermitian matrix is normal Unitary matrix is normal Hermitian unitary normal matrices Definition Let A be an n x n complex matrix 9 A is called Hermitian if A A o A is called unitary if AHA I o A is called normal if AHA AAH Hermitian matrix is normal Unitary matrix is normal Unitarily diagonalizable An n x n complex matrix A is normal if and only if there exists a unitary matrix U and a diagonal matrix Q such that U 1AU Q UHAU Hermitian and unitary matrix Let A be a Hermitian matrix 0 All the eigenvalues are real numbers 0 Eigenvectors belonging to distinct eigenvalues are orthogonal Hermitian and unitary matrix Let A be a Hermitian matrix 0 All the eigenvalues are real numbers 0 Eigenvectors belonging to distinct eigenvalues are orthogonal Let U be a unitary matrix 0 U 1 U o Eigenvectors belonging to distinct eigenvalues are orthogonal 0 Every eigenvalue ofA has ll 1 o For anyvectorv e C H Uv llll v H M Jordan decomposition Let A be an n x n complex matrix with detA7tl 71 t7A1mltiA2 quot2t7k quotk mt ti1 it72 2t7k k 39 For each find Jordan blocks J1 Jp where o the number of blocks is the geometric multiplicity of 0 at least one block has order r and all the others have order at most r o the sum of orders of the blocks is m o the number of Jordan blocks of possible orders is determined uniquely by A n deandecompos on Then there exists an invertible matrix S such that J11 J1P1 s4AS JkPk

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