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# Class Note for MATH 322 with Professor Glickenstein at UA

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Date Created: 02/06/15

Matrices and vectors Linear independence Vector space Rank Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Transposition Matrices and vectors Linear in 39 39 1 Matrices and vectors 0 An m X n matrix is an array with m rows and n columns It is typically written in the form all a12 a1n a21 a22 a2n A Z I 7 am1 am2 amn where i is the row index andj is the column index a 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 7 8 Linear Algebra Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Matrices a n ln Transposition Matrices and vectors continued 0 Examples 0 A 1 i J is a 2 X 2 square matrix with real entries 9 u 16 J is a column vector of A 0 0 3 7i complex entries 1 0 0 1 a BLO i 0 Jisa3gtlt3diagonal matrix with a An n X n diagonal matrix whose entries are all ones is called the n X n identity matrix 1 6 8 0 l 39S a 2 X 4 matrIX Wlth real entries Chapters 7 8 Linear Algebra C1 2 3 10 Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Transposition Matrices and vectors Linear indepe Vectc 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 ABltgtgtaijb forallij1 i m1 j n U7 0 The sum of A and B is the m x n matrix obtained by adding the entries of A to those of B Al B aij l bij o The product of A with the scalar c is the m x n matrix obtained by multiplying the entries of A by c CA caj Chapters 7 8 Linear Algebra Matrices and vectors Linear independence Vector Ra n k Mia n 0 Let A aij be an m x n matrix and B be an n x p matrix The product C A8 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 CU E aik bkj k1 1 2 1 2 3 10 ltgtExampesLetA 3 4andC1 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 Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Transposition Matrix multiplication continued o More examples 9 Consider the system of equations 3X1l 2X2 X3 4 X2 7X30 X1 l 4X2 6X3 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 0 Let Chapters 7 8 Linear Algebra Definitions Matrix addition and scalar multiplication 0 Matrix multiplication Rani Rules forimatrix addition and multiplication TranspQSItIon Matrices and vectors Linear indepe Vectc 3 Rules 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 if A and B are matrices and C1 and C2 are scalars then ABBA A l B l CA l B l C C1ABC1A l C1B C1C2A C1Al C2A C1 C2 A C1 C2A whenever the above quantities make sense Chapters 7 8 Linear Algebra Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Transposition Matrices a nd Linear indep Vect Rank Rules for matrix addition and multiplication continued 0 The product of two matrices is associative and distributive ie ABC ABC 2 ABC ABCABAC 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 ABy 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 8 32E 0 If A7 8 0 one says that the matrices A and B commute Chapters 7 8 Linear Algebra 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 aij then AT aji 12310 0 Example Find the transpose of C 1 6 8 0 0 Some properties of transposition If A and B are matrices and c is a scalar then ABTATBT CATCAT T A BT BTAT AT 2 A whenever the above quantities make sense Matrices and vectors Linear independence De nitions Vector E a i i R3 71 k in ience o A linear combination of the n vectors 31 32 3 is an expression of the form c131 l C232 l l CH3 where the C s are scalars 0 A set of vectors 31 32 3n 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 31 32 3n is linearly independent if and only if C131C232Cn3nOgtc1ZC2cn0 Matrices and w 39s Linear indepen Vector Evfmmioilag i Ra n k Limear i39h J i continued 0 Examples 0 Are the columns of the matrix A 16 i linearly independent 0 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 0 Can you find a condition on a set of n vectors which would guarantee that these vectors are linearly dependent Matrices and in Linear indepc Vector o A real or complex vector space is a non empty set V whose elements are called vectors and which is equipped with two operations called vector addition and multiplication by a scalar o The vector addition satisfies the following properties 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 a 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 3 E V a 0 a 9 For each a E V there exists a unique vector a E V such that a a 0 Matrices and in Linear indepc Vector Vector f o The multiplication by a scalar satisfies the following properties 0 The multiplication of a vector 3 E V by a scalar or E R or or E C is denoted by aa and is an element of V 9 Multiplication by a scalar is distributive aabaaab oz aoza a for all ab Vand 0566R or C It is associative a a or 6 a for all a E V and 056 6 R or C GD Multiplying a vector by 1 gives back that vector ie 1 a a for all a E V Matrices an Linear indepel Vector o The span of set of vectors U 31 32 an is the set of all linear combinations of vectors in M It is denoted by Spanal7 327 7an 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 Q SpanB S 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 T 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 IR Similarly the column space of A is the span of the column vectors of A and is a subspace of Rm The rank of a matrix A is the dimension of its column space Theorem The dimensions of the row and column spaces of a matrix A are the same They are equal to the rank of A Example Check that the row and column spaces of C 1 2 3 10 l 1 6 8 0 dimension are vector subspaces and find their h atnces and vectors Linearind c i Vector The wank tlhero em Rank 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 IR 0 The rank theorem states that if A is an m x n matrix then rankA l dim n 0 Example Find the rank and the null space of the matrix 12 3 10 Cl16 8 ol39 Check that the rank theorem applies

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