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# Class Note for MATH 410 with Professor Dostert at UA

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Date Created: 02/06/15

THE UNIVERSITY OF ARIZONA Math 410 Matrix Analysis Section 21 Real Vector Spaces Section 22 Subspaces Section 23 Span and Linear Independence Section 24 Bases and Dimension Paul Dostert January 15 2009 111 A Properties of a Vector Space A vector space is a set V equipped with 2 operations i Addition If 1110 E V then 11 w e V ii Scalar Multiplication If v e V and c e R then 011 e V and subject to the following axioms for all u v w e V and c d e R a Commutativity of v w w 11 b Associativity Of 2 u v w u v w c ldentity There isaOeVstv00vv d Inverse For each 11 e V El v e V st 11 v v v 0 e Distributivity c d v 07 dv 8 c v w C7 cw f Associativity of Scalar c dv ed 11 g Scalar Unity 1 e R is st 111 11 Ex Show that R is a vector space Ex Show that the set of polynomials of degree 3 n forms a vector space Ex Does the set of m gtlt n real matrices form a vector space Note We can generalize this to function spaces and real linear spaces section 71 If you are a mathematics major look at section 71 as well Ala Subspace A subspace of a vector space V is a subset W C V which is a vectors space under the same operations Prop A nonempty subset W C V of a vector space is a subspace iff i Foreveryv7w 6W vw e Wamp ii For every 11 E W and every 0 e R cv e W Ex Which of the following are subspaces a x7y7z R3xyz0 b 937y R293y 1 C mnxn7m RCabxcx2dx3a7b7c7d R d 2ab00 Ah Span Let v17 711k be in a vector space V A sum of the form k C1711 C2712 39 39 39 Ckvk E Civm z391 with 3239 scalars is known as a linear combination of the elements v17 71 Their span is the subset W span 1117 7 1 C V consisting of all possible linear combinations Prop The span W span 1117 7 1 of any finite collection of vector space elements forms a subspace of the underlying vectors space Ex Describe each of the below spaces Span1l7i 2 b span 1 0 7 0 0 A Elements in the Span Ex Decide if the given vector is in the span ltgtwlt gtyltzgtw lt2 Wm ME Emma C x x37spanx272xx27xx3 in 733 Ex Find a set that spans each of the given subspaces a The cz plane in R3 90 b y 3x2yz0 inR3 z c The set of 4th degree polynomials 734 A Linear Independence The vectors v17 711k 6 V are called linearly dependent if there are scalars 017 7 ck not all zero st 01111 Ck ljk 0 Vectors that are not linearly dependent are called linearly independent Ex Determine whether the following are linearly independent 1 a 1 7 0 1 b 1 7 0 and and l l ONODl l l Ol Ul llgt A Linear Independence Examples Ex Determine whether the following are linearly independent a x2714x2 C733 b 27 481D2 ac7 cos2 C 1 sinalc7 sin2x Thm Let v1 matrix vk e R and let A v1 vk be the corresponding n gtlt k a The vectors v1 7 11k 6 R are linearly dependent iff there is a non zero soln c 7 0 e R to Ac 0 b The vectors are linearly independent iff the only soln to Ac 0 is c 0 c A vector bis in the span of U1 solution 7vk iff Ac b has at least one Ex Determine whether the following are linearly independent by reducing a matrix of the vectors 1 and 1 3 5 A1 Linear Independence Properties Lemma Any collection of k gt n vectors in R is linearly dependent Note The converse is not necessarily true we can have two linearly dependent vectors in R717 n gt 2 Prop A set of k vectors in R is linearly independent iff the corresponding n gtlt k matrix A has rank k Prop A collection of k vectors spans R iff their n gtlt k matrix has rank n Note From the previous two theorems we can think of a rank of a matrix as representing the number of linearly independent rows Ex Is the following set of vectors linearly independent Why or why not 1 1 2 4 3 1 and 1 7 0 2 1 5 A7 Basis A basis of a vector space V is a finite collection of elements v17 that a spans V and b is linearly independent 7vn V We call the standard unit vectors e17 en e R the standard basis for Rquot 1 1 1 Ex Find a basis for span 2 7 1 7 2 0 2 8 1 2 4 Ex Find a basis for span 2 7 3 7 1 1 2 3 Thm Every basis of R consists of exactly n vectors A set of n vectors vn is nonsingular rankA n v177vn ER isabasisiffA v1 u Dimension Thm Suppose the vector space V has a basis v17 711 Then every other basis of V has the same number of elements in it This number is called the dimension of V and is written dimV n Lemma Suppose v17 7 Un span a vector space V Then every set of k gt n elements w17 710k 6 V is linearly dependent Pf Idea is that we write each 10239 as a linear combination of vj then show if Clwl Ckwk 0we can have07 0 The general idea here is that a basis for an n dimensional space must have n elements The very important lemma is Lemma The elements v17 711 form a basis of V iff every ac e V can be written uniquely as a linear combination of the basis elements TL 300101 39Cnvn E Civi 2391 Ex Find a basis for and the dimension of 732 A Matlab Examples Let us attempt to find a basis for the column space of 1 1 0 1 0 1 1 1 0 1 1 1 We start by entering A We then calculate the rref which actually returns 2 arguments the rref matrix R and the vector jb which gives the index of columns for a basis of A Le Ajb is a basis for the range of A To find the dimension we could also find the rank of A So in Matlab we could do A1 1 0101 1 101 1 1 R jb rrefA Ajb rankA Ajb would indicate that the 1522 and 4th columns form a basis and clearly the dimension is 3 Since there are 4 vectors and the dimension is 3 this would also indicate these vectors are linearly dependent

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