Linearly Independent Sets, Bases. Review. Remarks. A set of vectors,,, in a vector space is said to be linearly independent if the vector equation

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1 Liearly Idepedet Sets Bases p p c c p Review { v v vp} A set of vectors i a vector space is said to be liearly idepedet if the vector equatio cv + c v + + c has oly the trivial solutio = = { v v vp} The set is said to be liearly depedet if there exits weights c c c ot all p such that cv + c v + + c p p Remarks A set cotaiig the zero vector is liearly depedet A set of two vectors is liearly depedet if ad oly if oe is a multiple of the other

2 s Liearly Depededet Why? Cotais vector 6 9 Liearly Idepededet Why? Neither is multiple of other j ( j > ) j Theorem { v v vp} A idexed set of two or more vectors with v is liearly depedet if ad oly if some vector v vectors v v v is a liear combiatio of the precedig 5 = = = v v v Clearly v v ad v are liearly depedet. I fact v + v { } We see that v v v Spa v v { v v v } = { v v } Spa Spa 6

3 p = p + p Let be a set of vectors i P () () () where p t = t p t = t p t = t+ t Determie if this is a idepedet set. is a liearly depedet set {p p } is liearly idepedet but does ot spa P 7 8 Defiitio A basis set is a "efficiet" spaig set cotaiig o uecessary vectors A basis is a set of liearly idepedet vectors where each vector i the vector space ca be described as a liear combiatio of the basis vectors 9

4 Which sets form a basis? Defiitio Let H be a subspace of a vector space V { b b bp} A idexed set of vectors β = i V is a basis for H if i. β is a liearly idepedet set ad { b b b } ii. H = Spa p Remarks To show a fiite spaig set B is a basis for a vector space V We ca show The umber of vectors i the basis B equals the dimesio of V Ad either of the followig: The basis B is liearly idepedet or that the basis B spas V

5 Let be a set of vectors i P () () () where p t = p t = t p t = t Determie if this is a basis for P has the same dimesio just as P ( ) = α + β + γ Ay polyomial p t t t P ca be writte as a liear combiatio of as () p t = αp + β p + γ p Hece is a basis for P Practice Does form a basis for R There are two vectors the same size as R = There are two basic variables hece the vectors are idepedet Hece this is a basis for R Cosider the followig vectors i R e = e = e = These vectors form a matrix i echelo form hece they are liearly idepedet u ca be writte as a u Furthermore ay vector liear combiatio u= u u= ue + u e + + u e e e e are called the usual or stadard basis of R 5 5

6 Fid a basis for M = all matrices Basis of all matrices 6 Practice Describe vector space of upper triagular matrices 7 Fid a basis for P () t = all polyomials of degree { } The set S = t t t of + polyomials is a basis of P () t Ay polyomial is i Spa S. Also S is liearly idepedet. 8 6

7 Forms basis for R? [ v v v ] = Therefore v v v are liearly idepedet ad form a basis for R 9 Forms basis for R? No there are more vectors tha eteries per vector this set is liearly depedet ( ) Fid a basis for N A A= [ A ] = x = x x x5 x free x = 6x + 5x x = x + x 6 + x5 5 5 x free x5 free uvw is a spaig set for N ( A) uvw are liearly idepedet u v w Therefore { uvw } is a basis for N( A ) 7

8 Theorem 5 Spaig Set Thm { v v vp} { v v vp} Let S= be a set i V ad H =Spa a. If oe of the vectors i S - say v - is a liear combiatio of the remaiig vectors i S the the set formed from S by removig v still spas H k k {} b. If H some subset of S is a basis for H Fid a basis for R ( A) A = A = [ a a a a ] = = 6 [ b 8 Observe b b b] a = a ad b = b a = a+ 5 a ad b = b+ 5b Elemetary row operatios do ot affect the liear depedece amog the colums of the matrix Therefore Spa { a a a a} = Spa { a a} ad Spa a a is a basis for Col A { } Theorem 6 The pivot colums of a matrix A form a basis for Col A 8

9 Fid a basis for Spa v vv Let A = { v v } { v v v } Therefore Spa forms a basis for 5 Fid a basis for the hyperplae x + x + x + x = x = x x x x is free = x + x + x x is free x is free x Basis is Spa 6 Remarks A vector space ca have may differet bases Each of the bases will have the same umber of basis vectors The umber of basis vectors is called the dimesio For example: commo bases for R are rectagular cylidrical ad spherical; each usig three coordiates to spa R The coordiates may chage but the vector space does ot chage whe usig a differet base 7 9