### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Linear Algebra 1 MATH 322

GPA 3.58

### View Full Document

## 48

## 0

## Popular in Course

## Popular in Mathematics (M)

This 143 page Class Notes was uploaded by Vernie McCullough on Thursday October 15, 2015. The Class Notes belongs to MATH 322 at Millersville University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 48 views. For similar materials see /class/223533/math-322-millersville-university-of-pennsylvania in Mathematics (M) at Millersville University of Pennsylvania.

## Reviews for Linear Algebra 1

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/15/15

Real Vector Spaces MATH 322 LinearAgebral J Robert Buchanan Department of Mathematics Spring 2007 Introduction Almost all of the properties of R can be found in abstract vector spaces Today we will strip the properties of R to their most basic core and see that if other sets have these properties then they behave like abstract versions of R Vector Space Axioms Definition Let V be an arbitrary nonempty set on which two operations are defined addition if u and v are in V then u v e V u v is called the sum of u and v scalar multiplication if u e V and k is a scalar then ku e V ku is called the scalar multiple of u by k Definition V is a vector space ifthe following axioms hold for all u v and w in V and scalars k and l o u v e V closure under addition 9 u v v u commutativity of addition 0 u v w u v w associativity of addition 9 30 e V zerovectorst 0u u0 uVu e V 6 Vu e V 3 7 u e V negative of u st u7uiuu0 Q ku e V 0 kuvkukv 0 klu kulu 9 klu klu Q 1u u Verify that the following sets are vector spaces The scalars are real numbers 0R withn1 9 The set of all m x n matrices with real entries 9 The set of functions which are continuous on 01 9 The set of all infinite sequences which converge to O G The set 0 Verify that the following sets are vector spaces The scalars are real numbers 0R withn1 The set of all m x n matrices with real entries 9 The set of functions which are continuous on 01 9 The set of all infinite sequences which converge to O G The set 0 Remark the last example is called the zero vector space Nonvector space Example Let V R3 and define o uvU1V1U2V2usV3 o ku ku1kuz0 Question why is V not a vector space with respect to the vector addition and scalar multiplication Properties of Vectors Let V be a vector space and let u e V and k be a scalar then 0 Ou 0 9 k0 0 9 71u iu 0 ifku 0 thenk Ooru 0 ropes of Vectors 7 7 7 7 Let V be a vector space and let u e V and k be a scalar then 0 Ou 0 9 k0 0 9 71u iu 0 ifku 0 thenk Ooru 0 Homework Read Section 51 and work exercises 1 15 odd 21 Orthogonal Matrices Change of Basis MATH 322 LinearAgebral J Robert Buchanan Department of Mathematics Spring 2007 Sometimes choosing the appropriate basis when working with a vector space can make a specific problem easier Orthogonal Matrices Definition A square matrix A with the property that A 1 AT is said to be an orthogonal matrix Orthogonal Matrices Definition A square matrix A with the property that A 1 AT is said to be an orthogonal matrix Remark A square matrix A is orthogonal if and only if either AAT I orATA I Verify that A 7 71 J is an orthogonal matrix 3 2 4 N A N Equivalent Statements For an n x n matrix A the following statements are equivalent 0 A is orthogonal G The row vectors of A form an orthonormal set in R With the Euclidean inner product 9 The column vectors of A form an orthonormal set in R With the Euclidean inner product 7 quivalent Statements For an n x n matrix A the following statements are equivalent 0 A is orthogonal G The row vectors of A form an orthonormal set in R With the Euclidean inner product 9 The column vectors of A form an orthonormal set in R With the Euclidean inner product 1 ltgt 2 D Properties of Orthogonal Matrices Suppose that A and B are n x n orthogonal matrices then 0 A 1 is orthogonal 9 AB is orthogonal and 9 detA i1 thgal Matrices as Lr Operators For an n x n matrix A the following statements are equivalent 0 A is orthogonal 9 HAxH HxH foraIx e R 9 AxAyxyforaIxy ER rthogal Matrices as Linear Operators For an n x n matrix A the following statements are equivalent 0 A is orthogonal Q HAxH HxH foraIx e R 9 AxAyxyforaIxyeR 1 gt 2 D rthogal Matrices as Linear Operators For an n x n matrix A the following statements are equivalent 0 A is orthogonal Q HAxH HxH foraIx e R 9 AxAyxyforaIxyeR 1 gt 2 gt 3 D rthogal Matrices as Linear Operators For an n x n matrix A the following statements are equivalent 0 A is orthogonal Q HAxH HxH foraIx e R 9 AxAyxyforaIxyeR 12gt3gt1 D 7 rthognal Matrices as Linear Operators For an n x n matrix A the following statements are equivalent 0 A is orthogonal 9 llell llxll foraIx e R 9 AxAyxyforaIxyeR 12gt3gt1 D Remark If T R a R is multiplication by an orthogonal matrix A then T is called an orthogonal operator on R Coordinate Matrices 7 7 Recall If B V1V2 v is a basis for a vector space V and v e V then there are scalars k1 k2 k7 such that V k1V1k2V2 ann The vector of coordinates of v relative to B is denoted vg k17k2 k and the coordinate matrix of v relative to B is k1 k v15 2 kn Change of Basis Question If we change the basis for a vector space V from basis 3 to basis 8 how is Vg related to Vgl Change of Basis Question If we change the basis for a vector space V from basis 3 to basis 8 how is Vg related to Vgl Suppose V is ndimensional and B u1 U2 u and B V1V2 v are bases for V ChangeofBascon nuedi A Since 3 is a basis for V V1 P11U1P12U2p1nun V2 P21U1 P22U2 P2nUn V pn1u1pn2u2 Pnnun Let v e V then there exist scalars k1 k2 kn such that vhwbwmMM Change of Basis continued 7 i V k1V1k2V2ann k1P11U1P12U2 p1nun k2P21U1 P22U2 p2nun knpn1u1 pn2u2 Pnnun k1P11 k2P12 knP1nU1 k1P21 k2P22 kannU2 k1pn1 k2pn2 knpnnun Change of Basis continued 7 7 k1 k2 Thus if Vgl then kn P11 P12 P21 P22 MB Pn1 Pn2 Change of Basis continued 7 P11 P12 PM Let P p p pf PM Pn2 pnnj V118iv218i iVng P is called the transition matrix from B to B H w and find the transition matrix from B to B 1 O O 1 1 1 Properties of a Transition Matrix If P is the transition matrix from B to B then 0 P is invertible 9 P 1 is the transition matrix from B to B rope of Trnionatrix If P is the transition matrix from B to B then 0 P is invertible 9 P 1 is the transition matrix from B to B ropes of a Transitionatrix If P is the transition matrix from B to B then 0 P is invertible 9 P 1 is the transition matrix from B to B Vg PVg My P4 v15 Change of Orthonormal Basis If P is the transition matrix from one orthonormal basis to another orthonormal basis for an inner product space then P is an orthogonal matrix ie P 1 PT hang of Orthonormal Basis 7 7 If P is the transition matrix from one orthonormal basis to another orthonormal basis for an inner product space then P is an orthogonal matrix ie P 1 PT Homework Read Section 65 and work exercises 1 2 5 7 10 12 15 23 25 Linear Transformations from R to Rm MATH 322 LinearAgebral J Robert Buchanan Department of Mathematics Spring 2007 Introduction Today we will discuss the behavior and properties of functions whose domain is a subset ofthe vectors in R and whose range is a subset ofthe vectors in Rm Remark Linear transformations are of fundamental importance in science engineering and mathematics Functions Definition A function f is a rule of correspondence that associates to each element a e A the domain set a unique element b e B the codomain range faia e A Functions Definition A function f is a rule of correspondence that associat to each element a e A the domain set a unique element b e B the codomain range fala e A Remarks 0 Two functions f1 and f2 are equal if they have the same domain and if f1 3 f2a Va 6 A 0 When the domain is R and the codomain is Rm we call f a map or transformation from R to Rm f R a Rm is called an operator Generically a map f R a Rm can be expressed as W1 f1X17X277Xn W2 f2X17X277Xn Wm fmX17X27 7Xn Generically a map f R a Rm can be expressed as W1 f1X17X277Xn W2 f2X17X277Xn Wm fmx1X2 Xn If f1 f2 fm are linearfunctions of X1 X2 Xn then the transformation is called linear Matrix Multiplication 7 63911 63912 a1n X1 V1 63921 63922 a2n X2 Y2 8m1 8m2 amn Xn Vm Ax y If we name the transformation T R a Rm then A is said to be the standard matrix for the linear transformation T is multiplication by A If T R a Rm is such that Tx 0 for all x e R then Tx T0x Ox 0 is called the zero transformation If T R a Rm is such that Tx 0 for all x e R then Tx T0x Ox 0 is called the zero transformation If T R a R is such that Tx x for all x e R then Reflection Operators Suppose T lt Tlt X y H29 gt X y reflection about yaxis Reflection Operators X 71 O X Suppose T lt y gt O 1 y then T lt i gt reflection about yaxis X 1 O X Suppose T lt y gt O 71 then T lt X gt X reflection about Xaxis V iV Reflection Operators X 71 O X Suppose T lt y gt O 1 y then T lt i gt reflection about yaxis X 1 O X Suppose T lt y gt O 71 then reflection about Xaxis Piturs of Reflections Threedimensional Reflection 0 Reflection about xyplane X 1 O O X T y O 1 O y z 0 O 71 z 0 Reflection about xzplane Tililll Z1 ll2l 0 Reflection about yzplane Tililll E El lil Projection Operators Definition A projection operator sometimes called an orthogonal projection operator maps a vector to its orthogonal projection on a line or plane through the origin Projection Operators Definition A projection operator sometimes called an orthogonal projection operator maps a vector to its orthogonal projection on a line or plane through the origin 0 Projection onto the Xaxis T lt 0 Projection onto the yaxis T lt Pictures Project onto Xaxis Project onto Xaxis 1 O O O X X o o o 1 y m 1 1 D 5 x 5 07 x o 02 41 7F 1 r1 707srosro25 75 1 7025 025 705 705 075 Threedimensional Projection 0 Projection onto xyplane MW 3 Ei iii 0 Projection onto xzplane MW 5 ii iii 0 Projection onto yzplane O X 0 O X T y O 1 O y 1 z Rotation Operators Definition A operator that rotates a vector through a fixed angle 0 is called a rotation operator 7 Rotation perators Definition A operator that rotates a vector through a fixed angle 0 is called a rotation operator The standard matrix for the rotation operator on R2 Which rotates a vector counterclockwise through an angle 0 is cos 0 7 sin 0 sin 0 cos 0 finion A operator that rotates a vector through a fixed angle 0 is called a rotation operator The standard matrix for the rotation operator on R2 Which rotates a vector counterclockwise through an angle 0 is cos 0 7 sin 0 sin 0 cos 0 Example Suppose the vector x x52 12 is rotated 0 45 counterclockwise around the origin 371 cos 0 7 sIn 0 Q Q 1 1 Q W sIn 0 cos 0 2 1 1 x Dilation and Contraction Opeators Dilation and contraction operators stretch or compress vectors without rotation reflection or translation Tx kx where the action is o contraction if 0 g k lt 1 o dilation if 1 lt k Compositions of Linear Transfrmatios 7 7 Suppose TA R a Rk and TB Rk HR quot then TB 0 TA Rquot a Rm where TB 0 TAX TBTAX TBAX BAx BAX Compositions of Linear Transfrmations Suppose TA R a Rk and TB Rk HR quot then TB 0 TA Rquot a Rm where TB 0 TAX TBTAX TBAX BAx BAX Remarks 0 Composition of linear transformations is equivalent to multiplying the standard matrices representing the transformations TB 0 TA TBA 0 Since matrix multiplication is not commutative then composition of linear transformations is not commutative Rotate the vector x x52 12 by 0 37r4 counterclockwise around the origin and project it onto the Xaxis Homework Read Section 42 and work exercises 1 5 9 12 17 21 The Determinant Function MATH 322 LinearAgebral J Robert Buchanan Department of Mathematics Spring 2007 Basic Idea a determinant is a function with specific properties we will mention later whose domain is a set of square matrices and whose range is a set of scalar real numbers Introduction Basic Idea a determinant is a function with specific properties we will mention later whose domain is a set of square matrices and whose range is a set of scalar real numbers IfA a Z J then detA ad 7 be General Matrices Now we develop the notion of a determinant for n x n matrices Definition A permutation of the set of integers 12 n is any rearrangement without omissions or repetitions General Matrices Now we develop the notion of a determinant for n x n matrices Definition A permutation of the set of integers 12 n is any rearrangement without omissions or repetitions If we consider 123 there are 6 permutations 123 132 213 231 312 321 General Matrices Now we develop the notion of a determinant for n x n matrices Definition A permutation of the set of integers 12 n is any rearrangement without omissions or repetitions If we consider 123 there are 6 permutations 123 132 213 231 312 321 Remark There are n permutations of n objects Inversions Notation a permutation of 12 n will be written as 017127 7j 7 Any permutation can be expressed as the product of transpositions of pairs of integers Inversions Notation a permutation of 12 n will be written as 017127 7j 7 Any permutation can be expressed as the product of transpositions of pairs of integers Definition An inversion has occurred if a larger integer precedes a smaller integer Counting Inversions The total number of inversions in a permutation U1j2 jn can be determined as follows 0 Count the number of integers followingj1 that are smaller than j1 9 Add to that the number of integers followingjz that are smaller thanj2 9 Continue forj3j4 J39n71 Counting Inversions The total number of inversions in a permutation U1j2 jn can be determined as follows 0 Count the number of integers followingj1 that are smaller than j1 9 Add to that the number of integers followingjz that are smaller thanj2 9 Continue forj3j4 j1 Count the number of inversions in 432561 Definition A permutation is called even if the total number of inversions is even and is called oclcl if the total number of inversions is odd Definition A permutation is called even if the total number of inversions is even and is called oclcl if the total number of inversions is odd Determine the parities ofthe following two permutations 0 173747275 9 374757271 Defion Elementary Product An elementary prodiuct om an n x n ma x is any proctuct 7 of n entries from A such that no two factors come from the same row or column Elementary Product Definition An elementary product from an n x n matrix A is any product of n entries from A such that no two factors come from the same row or column List all the elementary products of the following matrices a a a 11 12 63921 63922 63911 63912 63913 9 63921 63922 63923 63931 63932 63933 Elementary Product Definition An elementary product from an n x n matrix A is any product of n entries from A such that no two factors come from the same row or column List all the elementary products of the following matrices 63911 63912 63921 63922 63911 63912 63913 9 63921 63922 63923 63931 63932 63933 Observation there are n elementary products of an n x n matrix Signed Elementary Products Think of an elementary product written as am 8212 anj where U1j2 j is a permutation ofthe set 12 n Definition A signed elementary product from A is an elementary product am 8212 anj multiplied by 0 1 if U1j2 j is an even permutation or o 71 if U1j2 j is an odd permutation Signed Elementary Products Think of an elementary product written as am 8212 anj where U1j2 j is a permutation ofthe set 12 n Definition A signed elementary product from A is an elementary product am 8212 anj multiplied by 0 1 if U1j2 j is an even permutation or o 71 if U1j2 j is an odd permutation List all the signed elementary products of the following matrices 63911 63912 63921 63922 l l Determinant Function Definition Let A be an n x n matrix the determinant of A denoted detA is the sum of all the signed elementary products ofA Determinant Function Definition Let A be an n x n matrix the determinant of A denoted detA is the sum of all the signed elementary products ofA Find the determinants of the following matrices 63911 0 63921 63911 9 63921 63931 63912 63922 63912 63922 63932 l 63913 63923 63933 Evaluating 3 x 3 Determinant 63911 63912 63913 63911 63912 63913 63911 63912 det 63921 63922 63923 63921 63922 63923 63921 63922 63931 63932 63933 63931 63932 63933 63931 63932 Add all the elementary products pointing to the right and subtract all the elementary products pointing to the left Evaluating 3 x 3 Determinant 63911 63912 63913 63911 63912 63913 63911 63912 det 63921 63922 63923 63921 63922 63923 63921 63922 63931 63932 63933 63931 63932 63933 63931 63932 Add all the elementary products pointing to the right and subtract all the elementary products pointing to the left Warning mnemonic device works only for 2 x 2 and 3 x 3 matrices Find the determinant of MatrixDeterminant Notation 63911 63912 63913 63911 63912 63913 det 63921 63922 63923 63921 63922 63923 63931 63932 63933 63931 63932 63933 Homework Read Section 21 and work exercises 1 13 and 17 21 odd Properties of the Determinant Function MATH 322 LinearAgebral J Robert Buchanan Department of Mathematics Spring 2007 Overview Today s discussion will illuminate some of the properties of the determinant 0 relationship with scalar products 0 multilinearity 0 relationship with matrix products 0 relationship with invertible matrices Basic Properties If A is an n x n matrix and k is a scalar then detkA k detA Bsic pes As If A is an n x n matrix and k is a scalar then detkA k detA A Nonproperty We can show by example that detA B 7 detA detB 1O 12 LetAO1andB3 4 detAB g 47 172detAdetB However If matrix A and B differ in just a single row LetA B and C be n x n matrices that differ only in a single row say the r row and suppose the r of C is the sum of the r rows ofA and B Then detC detA detB However If matrix A and B differ in just a single row LetA B and C be n x n matrices that differ only in a single row say the r row and suppose the r of C is the sum of the r rows ofA and B Then detC detA detB Remark the same result holds for columns However If matrix A and B differ in just a single row LetA B and C be n x n matrices that differ only in a single row say the r row and suppose the r of C is the sum of the r rows ofA and B Then detC detA detB Remark the same result holds for columns 63911 63912 b21 compare detA detB to LetA andB 63911 63912 a2 a2 and 63911 63912 63921 b21 63922 b22 etC Determinant of Matrix Product IfA andB are n x n matrices then detAB detA detB Determinant of Matrix Product IfA andB are n x n matrices then detAB detA detB J Remark we will first prove this for the case where A is an elementary matrix 7 eterminant of Matrix Product IfA andB are n x n matrices then detAB detA detB Remark we will first prove this for the case where A is an elementary matrix Suppose E is an elementary n x n matrix and B is an n x n matrix then detEB detE detB nt of Matr Prduct IfA antlB are n x n matrices then detAB detA detB Remark we will first prove this for the case where A is an elementary matrix Suppose E is an elementary n x n matrix and B is an n x n matrix then detEB detE detB er nt of Matrix Prduct IfA andB are n x n matrices then detAB detA detB Remark we will first prove this for the case where A is an elementary matrix Suppose E is an elementary n x n matrix and B is an n x n matrix then detEB detE detB detE1E2 EmB detE1 detE2 detEm detB 7 Ivertibility A square matrix A is invertible if and only if detA 7 O Inerti A A square matrix A is invertible if and only if detA 7 O lam DJ verti A square matrix A is invertible if and only if detA 7 O m Remark this is one of the most important results of linear algebra Determine if the matrix below is invertible 262 763 727 Rtu Previous Theor We can now prove the general case ofthe theorem IfA andB are n x n matrices then detAB detA detB etur Previous Theor 7 7 7 We can now prove the general case ofthe theorem IfA andB are n x n matrices then detAB detA detB Case A is singular eturnt Previous Theore 7 7 We can now prove the general case ofthe theorem IfA andB are n x n matrices then detAB detA detB Case A is singular Case A is invertible D 7 A Corollary If A is invertible then detA 1 detA 1 i 1 7 detA Aor A If A is invertible then detA 1 detA 1 i 1 7 detA Linear Systems Consider a linear system ofthe form Ax x where is a scalar Linear Systems 7 7 iii 7 7 Consider a linear system ofthe form Ax x where is a scalar Ax x Ax Ix Ix 7 Ax o I 7Ax o Linear Systems 7 7 7 7 Consider a linear system ofthe form Ax x where is a scalar Ax x Ax Ix Ix 7 Ax 0 I 7 Ax 0 If there exists a such that there is a nontrivial solution x then is called an eigenvalue and x is called an eigenvector Characteristic Equation 7 The linear system I 7 Ax 0 has a nontrivial solution when detI 7 A O The equation detI 7 A O is called the characteristic equation Characteristic Equation 7 The linear system I 7 Ax 0 has a nontrivial solution when detI 7 A O The equation detI 7 A O is called the characteristic equation Find the eigenvalues and eigenvectors of the following matrix l il Equivalent Statements If A is an n x n matrix then the following statements are equivalent 0 A is invertible G Ax 0 has only the trivial solution 0 The reduced row echelon form of A is In 9 A is expressible as the product of elementary matrices 9 Ax b is consistent for every n x 1 matrix b 9 Ax b has exactly one solution for every n x 1 matrix b e detA 7t 0 Homework Read Section 23 and work exercises 1a 2 4 5 11 odd 14a 15a16 1819 Further Results on Invertibility MATH 322 LinearAgebral J Robert Buchanan Department of Mathematics Spring 2007 Our discussion today will center on 0 more results related to inverses of matrices o more results related to solving systems of linear equations Recall our graphical argument that a linear system of two equations in two unknowns has either no solution a unique solution or infinitely many solutions Recall our graphical argument that a linear system of two equations in two unknowns has either no solution a unique solution or infinitely many solutions Every system of linear equations has either no solutions exactly one solution or in nitely many solutions Recall our graphical argument that a linear system of two equations in two unknowns has either no solution a unique solution or infinitely many solutions Every system of linear equations has either no solutions exactly one solution or in nitely many solutions Linear Systems If A is an invertible n x n matrix then for each n x 1 matrix b the system of equations Ax b has the unique solution x A 1 b nearstems 7 7 7 If A is an invertible n x n matrix then for each n x 1 matrix b the system of equations Ax b has the unique solution x A 1 b Using matrix inversion find the solution to the system of equations below X1 4X2 X3 0 X1 9X2 7 2X3 6X1 4X2 7 8X3 O A Common Coefficient Matrices i Often we encounter related linear systems where only the righthand side changes ie AXb17 AXb27 7 AXbk Common Coefficient Matrices Often we encounter related linear systems where only the righthand side changes ie AXb17 AXb27 7 AXbk There are two means of solving these systems of systems of equations Common Coefficient Matrices Often we encounter related linear systems where only the righthand side changes ie AXb17 AXb27 7 AXbk There are two means of solving these systems of systems of equations 0 Find A 1 and then calculate x A 1bfori12k 9 Form the augmented matrix Alb1lb2l lbk and use GaussJordan elimination Consider Ax b where 714 1 o 73 A 19 72 b1 1 and b2 4 6 4 78 1 75 Use GaussJordan elimination to solve both systems of equations Properties of Invertible Matrics Remark we defined the inverse of matrix A to be matrix B such that AB I and BA I Actually only one condition is needed Properties of Invertible Matrices Remark we defined the inverse of matrix A to be matrix B such that AB I and BA I Actually only one condition is needed LetA be an n x n matrix 0 IfB is an n x n matrix such that BA I then B A 1 G IfB is an n x n matrix such thatAB I then B A 1 7 roperties of Invertible Matrices Remark we defined the inverse of matrix A to be matrix B such that AB I and BA I Actually only one condition is needed LetA be an n x n matrix 0 IfB is an n x n matrix such that BA I then B A 1 G IfB is an n x n matrix such thatAB I then B A 1 More Equivalence Statements If A is an n x n matrix then the following statements are equivalent 0 A is invertible 9 Ax 0 has only the trivial solution 9 The reduced row echelon form of A is In 9 A is expressible as the product of elementary matrices Q Ax b is consistent for every n x 1 matrix b 6 Ax b has exactly one solution for every n x 1 matrix b More Equivalence Statements If A is an n x n matrix then the following statements are equivalent 0 A is invertible 9 Ax 0 has only the trivial solution 9 The reduced row echelon form of A is In 0 A is expressible as the product of elementary matrices 6 Ax b is consistent for every n x 1 matrix b 9 Ax b has exactly one solution for every n x 1 matrix b iPrevioust we showed 1 gt 2 gt 3 gt 4 gt 1 More Equivalence Statements If A is an n x n matrix then the following statements are equivalent 0 A is invertible G Ax 0 has only the trivial solution 9 The reduced row echelon form of A is In 0 A is expressible as the product of elementary matrices Q Ax b is consistent for every n x 1 matrix b 6 Ax b has exactly one solution for every n x 1 matrix b More Equivalence Statements If A is an n x n matrix then the following statements are equivalent 0 A is invertible G Ax 0 has only the trivial solution 9 The reduced row echelon form of A is In 0 A is expressible as the product of elementary matrices Q Ax b is consistent for every n x 1 matrix b 6 Ax b has exactly one solution for every n x 1 matrix b 7 ore Equivalence Statements If A is an n x n matrix then the following statements are equivalent 0 A is invertible G Ax 0 has only the trivial solution 9 The reduced row echelon form of A is In 0 A is expressible as the product of elementary matrices Q Ax b is consistent for every n x 1 matrix b 6 Ax b has exactly one solution for every n x 1 matrix b Invertible Matrix Factors Factrs Recall we have seen that if n x n matrices A and B are invertible then AB is invertible Invertible Matrix Factors Factors Recall we have seen that if n x n matrices A and B are invertible then AB is invertible The converse is also true LetA and B be n x n matrices If AB is invertible then A and B are invertible Ivertie Matrix Factors Factors Recall we have seen that if n x n matrices A and B are invertible then AB is invertible The converse is also true LetA and B be n x n matrices If AB is invertible then A and B are invertible Fundamental Problem 7 Problem let A be a fixed m x n matrix Find all the m x 1 matrices b such that the system of equations Ax b is consistent Determine by rowreducing the augmented matrix the conditions which b1 b2 and b3 must satisfy for the following system to be consistent X1 2X27X3 b1 4X1 5X2 7 X3 b2 4X1 7X2 ng b3 Homework Read Section 16 and work exercises 1 5 9 16 21 24

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made $280 on my first study guide!"

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.