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# Linear Algebra With Applications MA 51100

Purdue

GPA 3.97

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This 48 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 51100 at Purdue University taught by Staff in Fall. Since its upload, it has received 66 views. For similar materials see /class/208127/ma-51100-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

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 Hm it 316 m 1f vf wmmx A 0 Find the RREF of the matrix A In C11 C21 C31 C12 C22 C32 C13 C23 C33 3 72 1 LetA 5 6 2 ThenA 194 1 o 73 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 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 0 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 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 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 0 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 ll 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 o Tt1t2t11t2 t717 t1 1 0 Tt71t27t711t21 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 a Tt1t2t11t211t71 o Tt71t27t711t21 1 12 J7 Tt1lg 12 1 1 o The representation is 12 12 O 15 1 2 732 n Standard inner product The following are standard inner products 0 In R lt v W gt VTW o In C lt vw gt VHW o In C01lt fg gt f01fg 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 PrOJWV77W177W2quot39 T Wd 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 77W1 Tiwz W T Wd 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 7 1 7 2 W3 7 V3 WW1 W1 wgm W2 wTv WT Vm 7 7 1 m 7 7 L Wm 7 Vm Maw1 W1 WILWWIWW Wm71 o W17W2 Wm are orthogonal 1 1 1 o 7W 7W W are orthonormal HW1H 1 HW2H 2 WmH m Eigenvalues eigenspace and multiplicities Let A be an n x n matrix a 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 a 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 a 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 71 7 1 quot1 7 2 quot2 7 Ak 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 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 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 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 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 any vector v e C H Uv llll v H M Jordan decomposition Let A be an n x n complex matrix with detA 7 tln 71 t 7 A1 quot1 t 7 2 quot2 t 7 AK mt t 7 A1 t 7 2 2 t 7 AK 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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