Linear Algebra MATH 300
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Date Created: 10/23/15
Math 300 Notes for Section 61 1 De nition ofa Linear Transformation If T V gt W is a function from a vector space V to a vector space W then T is called a linear transformation from V to W if the following two conditions are satis ed for all vectors 1 and 17 in V and for every scalar c a TO 1 5 TO T07 T preserves addition b Tc CTOZ T preserves scalar multiplication 2 Theorem Properties ofLinear Transformations If T V gt W is alinear transformation then a T6 6 b T U TU for every 1 in V c TO 1 5 T1Z T07 for all 1 and 17 in V 3 Theorem Linear Transformations Preserve Linear Combinations If T V gt W is alinear transformation then T61171 62172 ck k clT171 62T172 CkTGk for all vectors 171 172 17k in V and all scalars 6162 ck 4 Theorem Linear Transformation given by a Matrix Let A be an m X n matrix Then the function T R gt Rm de ned by T 17 A17 is a linear transformation Math 300 Notes for Section 12 l Matrices in Row Echelon Form A matrix is said to be in row echelon form if it has the following properties a All rows with all entries 0 are at the bottom of the matrix b The rst nonzero number in any nonzero row is a 1 called a leading l c In any two successive nonzero rows the leading l in the lower row occurs farther to the right than the leading l in the higher row If in addition each column that contains a leading 1 has zeros everywhere else we say that the matriX is in reduced row echelonform 2 Elementary Row Operations The following operations performed on the rows of a matriX are called elementary row operations a Interchange two rows b Multiply a row by a nonzero scalar c Add a multiple of one row to another 3 Gauss Jordan Elimination The Gauss Jordan elimination procedure for solving a linear sys tem a Form the augmented matriX of the system b Transform the augmented matriX into reduced rowechelon form c Write down the linear system that corresponds to the matriX in rowreduced echelon form This system has the same solution set as the original system d Solve each equation in the system obtained in c for the leading variable the variable cor responding to the leading l e Assign arbitrary values 3 i to the free variables the nonleading variables if any Homogeneous Systems Every homogeneous system a system that has zeroes on the right is 4 consistent since x1 0 x2 0 xn 0 is always a solution Ifa homogeneous system has fewer equations than variables then it has in nitely many solutions since some variables have to be free 5 Derive Tips a The following functions perform elementary row operations multA i s Multiplies row i of matriX A by scalar s swapA i j Interchanges rows i and j of matriX A addA s i j Adds s times row i ofmatriX A to row j To use mult swap or add you need to load the le m300 b The following function gives the rowreduced echelon form of a matriX A rowreduce A Math 300 H to 03 F U a Notes for Section 11 Linear Equations A linear equation in n variables 21 2 can be written in the standard form mn is an equation that a1901 a22ann 57 where the coef cients a1 a2 an and the constant I are real numbers Solutions A solution to a linear equation in n variables is a sequence of it real num bers 31 32 3 such that the equation is satis ed when we substitute 1 31 2 32mn 3 Systems of linear equations A system ofm linear equations in n variables is a set of in linear equations in n variables a11961 112952 1mm b1 121961 122952 1mm 52 M1901 am z amng n bm A solution to such a system is a sequence of it real numbers 31 32 3 that is simulta neously a solution for each equation in the system To solve a linear system is to nd the set of all solutions to the system Method of Elimination To solve a system of linear equations we use the method of elimination which consists of repeatedly performing the following operations a Interchange two equations b Multiply an equation by a nonzero constant A c Add a multiple of one equation to another equation Theorem Number of Solutions A linear system may have no solutions exactly one solution or in nitely many solutions If it has no solutions we say that it is inconsistent If it has at least one solution we say that it is consistent Derive Tip To plot several planes say my 2 3 my 72 1 and 71 2 2 rst solve each equation for 2 2 7m 7 y 3 2 z y 71 and 2 7m y 2 Enter the expression max7m 7 y 3 z y 7 1 7m y 2 and plot it in 3D plot window Math 300 N E V39 9 gt1 gt0 Notes for Section 62 Kernel of a Linear Transformation If T V gt W is a linear transformation then the kernel of T denoted by kerT is the set of all vectors 17 in V such that T17 0 Theorem Kernel is a Subspace of V The kernel of a linear transformation T V gt W is a subspace of the domain V Nullity of T If T V gt W is a linear transformation then the dimension of kerT is called the nullity of T Range of a Linear Transformation If T V gt W is a linear transformation then the range of T denoted by ranT is the set of all vectors 117 E W such that 117 T17 for some 17 e V Theorem Range is a Subspace of W The range of a linear transformation T V gt W is a subspace of the codomain W Rank of T If T V gt W is a linear transformation then the dimension of ranT is called the rank of T Linear Transformation and its Matrix Representation Let T R gt Rm be a linear transformation and let A be the matrix that represents T T 17 A17 Then kerT solution space of A 6 ranT column space ofA rank of T rank of A One to one Linear Transformations A linear transformation T V gt W is called one t0 one if for all 17 and 17 in V ifT17 T17then17 17 Theorem Relation between kerT and T being one to one Let T V gt W be alinear transformation Then T is onetoone if and only if kerT 0 Onto Linear Transformations A linear transformation T V gt W is called onto if for every 117 E W there is 17 E V such that 117 T17 Dimension Theorem Let V and W be nite dimensional vector spaces If T V gt W is a linear transformation then dimkerT dimranT dimV Math 300 Notes for Section 21 l Matrices An m X rt matrix A is a rectangular array of numbers arranged in m horizontal rows and rt vertical columns all 112 am 121 122 an A aml am am We say that A is of size m by n The number aij in the ith row and jth column ofA is called the i j entry of A and we write A aij Ifm n we say that A is a square matrix of order n 2 Equality of Matrices Two matrices are said to be equal if they have the same size and their corresponding entries are equal 3 Matrix Addition and Subtraction IfA aij and B blj are matrices ofthe same size then the sum A B and ali erertee A B are de ned by A B aij bij add corresponding entries A B aij bij subtract corresponding entries Matrices of different sizes cannot be added or subtracted 4 Scalar Multiplication IfA aij is a matrix and c is a scalar then the scalar multiple CA is de ned by CA calj multiply each entry by c 5 Matrix Multiplication IfA aij is an m X rt matrix and B blj is a rt X 1 matrix then the product AB is the m X 1 matrix whose entries are determined as follows To nd the entry in the row i and column j of AB a Single out row i from A and column j from B b Multiply the corresponding entries from the row and column together and then add up the resulting products Symbolically AB clj where n Cij ai1b1j ai2b2j quot39ainbnj Zaikbkf k1 6 Matrix Form of a Linear System Consider a linear system with m equations and n unknowns a11x1 aux 11an b1 a21x1 azzxz 12an b2 I 1 amlxl amzxz a an bm De ne the following matrices all 112 11 X1 b1 121 122 12 x2 b2 A x b aml amZ amn xn bm The matrix A is called the coef cient matrix of the linear system 1 Verify that 7 all 112 11 HVX1 7 a11x1 aux 11an 121 122 12 X2 a21x1 azzxz 12an A 1 hing any x amixl am xz am xn Therefore the linear system 1 can be written as a single matrix equation A 5 matrix form of a linear system Let 51 3 be the columns of A Then A x151 x221 2 xnz39in Hence the system A b is consistent if and only if b is a linear combination of the columns ofA Math 300 Equot E 3 V39 9 Theorem Properties of Matrix Addition Notes for Section 22 If A B and C are m X n matrices then a ABCABC bABBA Theorem Additive Identity and Inverses Let Omn be the m Xn matrix Whose each entry is 0 LetA allf be any m X n matrix and let A alf Then a A0mn Aand0mnAA b A A Omn and A A Om Theorem Properties of Matrix Multiplication If A B and C are matrices with sizes such that the given matrix operations are de ned then a ABC ABC b AB C AB AC c A BC AC BC Theorem Properties ofthe Identity Matrix IfA is an m X n matrix then a A1 A b ImA A It follows that if A is a square matrix of order n then InA A1 A Theorem Properties of Scalar Multiplication If A B and C are matrices with sizes such that the given matrix operations are de ned and C and d are scalars then a CdA CdA b 1A A c CA B CA CB d C dA CA dA e CAB CAB ACB Theorem Properties of Transposes If A and B are matrices with sizes such that the given matrix operations are de ned and C is a scalar then a ATT A b A BT AT BT c CAT CAT d ABT BTAT Math 300 1 N E Finding the Eigenvalues ofA Notes for Section 71 Eigenvalues and Eigenvectors ofa Matrix Let A be an n X n matrix A scalar A is called an eigenvalue of A if there is a nonzero vector 1 in R such that A1 A1 The vector 1 is called an eigenvector of A corresponding to A Note that A may be 0 Let A be an n X n matrix To nd the eigenvalues of A compute the determinant detAI A and solve the equation detAI A 0 The real solutions of the equation are the eigenvalues of A When expanded the determinant detAI A is a polynomial in A of degree n called the characteristic polynomial of A The equation detAI A 0 is called the characteristic equation of A Finding the Eigenvectors Corresponding to A Let A be an eigenvalue ofa matrix A To nd the eigenvectors corresponding to A form the matrix AI A and solve the homogeneous system AI A 6 The nonzero solutions are the eigenvectors corresponding to A The set of Q solutions the eigen vectors plus the zero vector is called the eigenspace of A The eigenspace as the solution space of a homogeneous system is a subspace of R Math 300 N E 4 U 6 Theorem Uniqueness of Inverse Method for Finding A l Theorem Inverse ofa Product Notes for Section 23 Inverse of a Matrix Let A be a square matrix of size n X n If there is a matrix B of the same size such that AB In and BA In then A is said to be nonsingular or invertible and B is called an inverse of A Square matrices that do not have an inverse are called singular If B and C are both inverses of a matrix A then B C Notation A71 The unique inverse of a nonsingular matrix A is denoted by A 1 We have AA 1 1 and A lA In Let A be an n X n matrix a Form the matrix A In b Transform A In to reduced rowechelon form c If A can be reduced to In then the matrix obtained in b has the form In A l If A cannot be reduced to I n then A is singular Suppose that A and B are nonsingular matrices of size n X n Then the product AB is nonsingular and AB 1 B IA I Method of Inverses Consider a linear system A B where A is a square matrix If A is nonsingular we can multiply both sides on the left by A 1 obtaining the unique solution i A 113 Math 300 Notes for Section 42 1 De nition of Vector Space Let V be a set on which two operations vector addition and scalar multiplication are de ned If the following axioms are satis ed for all 17 17 and 117 in V and all scalars real numbers 6 and d then V is called a vector space and the elements of V are called vectors Addition a 17 17 is avector in V V is closed under addition b 17 17 17 17 Vector additionis commutative c 17 17 17 17 17 117 Vector additionis associative d V has an additive identity that is there is a vector 3 in V such that for all 17 in V 17 17 and 31717 e Every vector in V has an additive inverse that is for every 17 in V there is a vector 17 in V such that 1717 and 1717 where Z is the additive identity from e Scalar Multiplication 1 C17 is a vector in V V is closed under scalar multiplication g c17 17 617 617 Distributive Property h c d17 617 d17 Distributive Property i Cd17 cd17 Multiplication by scalars is associative j 117 17 l is the identity for scalar multiplication 2 Standard Examples of Vector Spaces a The set R of all rivectors with the usual operations of addition and scalar multiplication de ned for rivectors b The set Mm of all m X n matrices with the operations of matrix addition and scalar multi plication c The set C 00 00 of all continuous functions de ned on the real line with the operations f gx fx gx Cfx Cfx d The set P of all polynomials with the operations as in c e The set P of all polynomials of degree 5 n with the operations as in c 3 Uniqueness of Additive Identity and Additive Inverses If V is a vector space then a V has exactly one identity We call this unique additive identity the zero vector and denote it by 0 Thus for every 1 in V 56a and61211 b Every vector 1 in V has exactly one additive inverse We denote this unique additive inverse of by 12 Thus 12 120 and 1170 4 Theorem Properties of Scalar Multiplication If V is a vector space 1 is a vector in V and c is a scalar then a 012 6 b c6 6 c Ifcil athenc OorlZ d 11Z 17 Math 300 Equot E 5 V39 6 Theorem Basis Test When We Know That dimV n Notes for Section 45 De nition ofa Basis Let V be a vector space A set S 131 1 52 called a basis for V if the following conditions hold 1n ofvectors in V is a S spans V b S is linearly independent Theorem Spanning and Linear Dependence If a set S 131132 every set T containing more than n vectors in V is linearly dependent Bn spans V then All Bases for V Have the Same Size has n vectors If V has a basis with n vectors then every basis for V De nition of Dimension Let V be a vector space with a basis consisting of n vectors We call the number n the dimension of V and write dimV n If V 0 the dimension of V is de ned to be zero Finite Dimensional and In nite DimensionalVector Spaces Not every vector space V has a nite basis If V has a nite basis or V 0 then V is called nite dimensional Otherwise V is called in nite dimensional Suppose V is a vector space with dimV n and let S 131132 1kbe a set ofk vectors in V Then a Ifk lt n then S does not span V b If k gt n then S is linearly dependent c If S is linearly independent andk n then S spans V d If S spans V andk n then S is linearly independent
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