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# Class Note for MATH 322 with Professor Glickenstein at UA

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Date Created: 02/06/15

Chapters 7 8 Linear Algebra Sections 71 72 amp 74 Chapters 78 Linear Algebra Matrices and vectors gt 1 and scalar Inultlpl Linear m tion 1 Matri and vectors 0 An m x n matrix is an array with m rows and n columns It is typically written in the form 81 a12 39 39 39 aln a2 322 39 39 39 32n A Z Z 7 3m1 3m2 amn where i is the row index and j is the column index 0 A column vector is an m x 1 matrix Similarly a row vector is a 1 x n matrix 0 The entries aij of a matrix A may be real or complex Chapters 78 Linear Algebra Matrices a nd vectors LINE Transposit Matres and vectors continued 0 Examples 0 A i i J is a 2 X 2 square matrix with real entries 0 u i J is a column vector of A 1 0 0 o B 0 i 0 is a 3 X 3 diagonal matrix with 0 0 3 7i complex entries An n X n diagonal matrix whose entries are all ones is called the n X n identity matrix 12310 16 80 Chapters 78 Linear Algebra is a 2 X 4 matrix with real entries C Definitions Matr addition and scalar multiplication Matt multiplication R rix addition and multiplication Matri s and vec 5 Linear l Matrix addition and scalar multiplication Let A aij and B be two m X n matrices and let c be a scalar o The matrices A and B are equal if and only if they have the same entries ABltgtajbj1 orallij7 lgigm7 lgjgn o The sum of A and B is the m X n matrix obtained by adding the entries of A to those of B ABajbj 0 The product of A with the scalar c is the m X n matrix obtained by multiplying the entries of A by c CA caij Chapters 78 Linear Algebra 39 39ons addition and scalar multiplication multiplication 1atrix addition and multiplication Transposition Matrices a nd vectors Line 2 Matrix multiplication 0 Let A aij be an m X n matrix and B be an n X p matrix The product C 2 AB of A and B is an m X p matrix whose entries are obtained by multiplying each row of A with each column of B as follows n C0 Z aik bkj k1 12 12 310 oExamplesLetA3 4andC l 6 8 0 o Is the product AC defined If so evaluate it 0 Same question with the product CA 0 What is the product of A with the third column vector of C Chapters 78 Linear Algebra Matri s and vec 5 Linear l Matrix multiplication continued a More examples 0 Consider the system of equations 3X1 2X2 X34 X2 7X3 0 X1 4X2 6X3 2 10 Write this system in the form AX Y where A is a matrix and X and Y are two column vectors 12 56 A3 4 and B7 8 Calculate the products AB and BA 6 Let Chapters 78 Linear Algebra Definitions 3 Res for matrix addition and multiplication o The rules for matrix addition and multiplication by a scalar are the same as the rules for addition and multiplication of real or complex numbers 0 In particular ifA and B are matrices and C1 and C2 are scalars then ABBA Al Bl CA l BC C1ABC1AC1B C1C2A C1Al C2A C1 C2A C1 C2A whenever the above quantities make sense Chapters 78 Linear Algebra Matrices a nd vectors Transposition Rules for matrix addition and multiplication continued o The product of two matrices is associative and distributive ie ABC ABC ABC AB l CAB l AC A l BCAC l BC 0 However the product of two matrices is not commutative If A and B are two square matrices we typically have ABg BA o For two square matrices A and B the commutator of A and B is defined as A B 2 AB BA In general A7 B 75 0 If A7 B 0 one says that the matrices A and B commute Chapters 78 Linear Algebra Definitions addition and scalar multiplication on Rules for mat clition and multiplication Transposition Matrices and vectors Linear in I 4 Transposition o The transpose of an m x n matrix A is the n x m matrix AT obtained from A by switching its rows and columns ie if A aij7 then AT 2 aJi 12310 16 8039 0 Example Find the transpose of C 0 Some properties of transposition If A and B are matrices and c is a scalar then A BT 2 AT BT CAT 2 CAT T A BT BTAT AT 2 A whenever the above quantities make sense Chapters 78 Linear Algebra Definitions Examples 5 Linear independence 0 A linear combination of the n vectors al 22 an is an expression of the form C181 C282 Cnam where the c s are scalars o A set of vectors 21 22 an is linearly independent if the only way of having a linear combination of these vectors equal to zero is by choosing all of the coefficients equal to zero In other words 21 22 7a is linearly independent if and only if C181C282Cnan0gtC1C2CnO Chapters 78 Linear Algebra s a nd vectors ndependence Definitions 5 Examples Linear independence 0 Examples 0 Are the columns of the matrix A i i linearly independent o Same question with the columns of the matrix C 1 2 3 10 1 6 8 0 39 0 Same question with the rows of the matrix C defined above 0 A set that is not linearly independent is called linearly dependent a Can you find a condition on a set of n vectors which would guarantee that these vectors are linearly dependent Chapters 78 Linear Algebra Definitions Bases and dimension 6 Vector space 0 A real or complex vector space is a nonempty set V whose elements are called vectors and which is equipped with two operations called vector addition and multiplication by a scalar 0 The vector addition satisfies the following properties 0 The sum of two vectors a E V and b E V is denoted by a b and is an element of V It is commutative 2 b b a for all a b e V It is associative a b c a b c for all a b c e V There exists a unique zero vector denoted by 0 such that for every vector 2 E V a 0 a O 990 For each a E V there exists a unique vector 2 E V such that a a 0 Chapters 78 Linear Algebra Definitions Vector space Bases and dimension Rank Vect space continued 0 The multiplication by a scalar satisfies the following properties 0 The multiplication of a vector 2 E V by a scalar oz 6 R or a E C is denoted by aa and is an element of V 9 Multiplication by a scalar is distributive ozabozaozb7 oz aoza a7 for all ab 6 V and 045 6 R or C 9 It is associative a a a5 a for all a E V and a 6 R or C 0 Multiplying a vector by 1 gives back that vector ie 1 a 37 for all a E V Chapters 78 Linear Algebra Linear ind Definitions Bases and dimension Bases and dimension 0 The span of set of vectors L1 317 22 7a is the set of all linear combinations of vectors in L1 It is denoted by Spanal7 327 39 39 39 73 or and is a subspace of V o A basis 8 of a subspace S of V is a set of vectors of S such that 0 SpanB 5 9 B is a linearly independent set 0 Theorem If a basis 8 of a subspace S of V has n vectors then all other bases of 5 have exactly n vectors 0 The dimension of a vector space V or of a subspace S of V spanned by a finite number of vectors is the number of vectors in any of its bases Chapters 78 Linear Algebra Definitions T he ran k theorem o The row space of an m x n matrix A is the span of the row vectors of A If A has real entries the row space of A is a subspace of R 0 Similarly the column space of A is the span of the column vectors of A and is a subspace of Rm 0 The rank of a matrix A is the dimension of its column space 0 Theorem The dimensions of the row and column spaces of a matrix A are the same They are equal to the rank of A 0 Example Check that the row and column spaces of 1 2 3 10 C 1 6 8 O J are vector subspaces and find their dimension Chapters 78 Linear Algebra Met Linear Definitions The rank theorem The rank theorem o The null space of an m x n matrix A NA is the set of vectors u such that Au 2 0 If A has real entries then NA is a subspace of R o The rank theorem states that if A is an m x n matrix then rankA dim n 0 Example Find the rank and the null space of the matrix C 1 2 3 10 1 6 8 O 39 Check that the rank theorem applies Chapters 78 Linear Algebra

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