Math 2331 Linear Algebra - 1.7 Linear Independence

1 Linear IndependenceMath 2331 Linear Linear IndependenceJiwen HeDepartment of Mathematics, university of jiwenhe/math2331 Jiwen He, university of HoustonMath 2331, Linear Algebra1 / Linear IndependenceDefinition Matrix Columns Special Linear IndependenceLinear Independence and Homogeneous SystemLinear Independence : DefinitionLinear Independence of Matrix ColumnsSpecial CasesA Set of One VectorA Set of Two VectorsA Set Containing the0 VectorA Set Containing Too Many VectorsCharacterization of Linearly Dependent SetsTheorem: Linear Dependence and Linear CombinationJiwen He, university of HoustonMath 2331, Linear Algebra2 / Linear IndependenceDefinition Matrix Columns Special CasesLinear Independence and Homogeneous SystemExampleA homogeneous system such as 1 2 33 595 93 x1x2x3 = 000 can be viewed as a vector equationx1 135 +x2 259 +x3 393 = 000.

2 The vector equation has the trivial solution (x1= 0,x2= 0,x3= 0), but is this theonly solution?Jiwen He, university of HoustonMath 2331, Linear Algebra3 / Linear IndependenceDefinition Matrix Columns Special CasesLinear Independence : DefinitionLinear IndependenceA set of vectors{v1,v2,..,vp}inRnis said to belinearlyindependentif the vector equationx1v1+x2v2+ +xpvp=0has only the trivial DpendenceThe set{v1,v2,..,vp}is said to belinearly dependentif thereexists weightsc1,..,cp,not all 0, such thatc1v1+c2v2+ +cpvp=0. Linear dependence relation(when weights are not all zero)Jiwen He, university of HoustonMath 2331, Linear Algebra4 / Linear IndependenceDefinition Matrix Columns Special CasesLinear Independence : ExampleExampleLetv1= 135 ,v2= 259 ,v3= 393.

3 A. Determine if{v1,v2,v3}is linearly If possible, find a Linear dependence relation amongv1,v2, : (a)x1 135 +x2 259 +x3 393 = 000 .Augmented matrix: 1 2 3 03 59 05 93 0 12 3 00 118 00 118 0 12 3 00 118 0000 0 x3is a free variable there are nontrivial He, university of HoustonMath 2331, Linear Algebra5 / Linear IndependenceDefinition Matrix Columns Special CasesLinear Independence : Example (cont.) {v1,v2,v3}is a linearly dependent set(b) Reduced echelon form: 1 03300 1 18 00 000 = x1=x2=x3 Letx3=(any nonzero number).Thenx1=andx2=. 135 + 259 + 393 = 000 orv1+v2+v3=0(one possible Linear dependence relation)Jiwen He, university of HoustonMath 2331, Linear Algebra6 / Linear IndependenceDefinition Matrix Columns Special CasesLinear Independence of Matrix ColumnsExample ( Linear Dependence Relation) 33 135 + 18 259 + 1 393 = 000 can be written as the matrix equation: 1 2 33 595 93 33181 = 000.

4 Each Linear dependence relation among the columns ofAcorresponds to a nontrivial solution toAx= columns of matrixAare linearly independent if and only if theequationAx=0hasonlythe trivial He, university of HoustonMath 2331, Linear Algebra7 / Linear IndependenceDefinition Matrix Columns Special CasesSpecial Cases: 1. A Set of One VectorSometimes we can determine Linear Independence of a set withminimal (1. A Set of One Vector)Consider the set containing one nonzero vector:{v1}The only solution tox1v1= 0 isx1=.So{v1}is linearly independent whenv16= He, university of HoustonMath 2331, Linear Algebra8 / Linear IndependenceDefinition Matrix Columns Special CasesSpecial Cases: 2.

5 A Set of Two VectorsExample (2. A Set of Two Vectors)Letu1=[21],u2=[42],v1=[21],v2=[2 3].a. Determine if{u1,u2}is a linearly dependent set or a linearlyindependent Determine if{v1,v2}is a linearly dependent set or a linearlyindependent :(a) Notice thatu2=u1. Thereforeu1+u2= 0 This means that{u1,u2}is a He, university of HoustonMath 2331, Linear Algebra9 / Linear IndependenceDefinition Matrix Columns Special CasesSpecial Cases: 2. A Set of Two Vectors (cont.)(b) Supposecv1+dv2= 0. But this is impossible sincev1isa multiple ofv2which meansc=.Similarly,v2=v1ifd6= this is impossible sincev2is not a multiple ofv1and sod= means that{v1,v2}is a He, university of HoustonMath 2331, Linear Algebra10 / Linear IndependenceDefinition Matrix Columns Special CasesSpecial Cases: 2.

6 A Set of Two Vectors (cont.)A set of two vectors is linearly dependent if at least one vector is amultiple of the set of two vectors is linearly independent if and only if neither ofthe vectors is a multiple of the He, university of HoustonMath 2331, Linear Algebra11 / Linear IndependenceDefinition Matrix Columns Special CasesSpecial Cases: 3. A Set Containing the0 VectorTheoremA set of vectorsS={v1,v2,..,vp}inRncontaining the zerovector is linearly :Renumber the vectors so thatv1=. Thenv1+v2+ +vp=0which shows thatSis He, university of HoustonMath 2331, Linear Algebra12 / Linear IndependenceDefinition Matrix Columns Special CasesSpecial Cases: 4.

7 A Set Containing Too Many VectorsTheoremIf a set contains more vectors than there are entries in each vector,then the set is linearly dependent. any set{v1,v2,..,vp}inRnis linearly dependent ifp> of Proof:A=[v1v2 vp]isn pSupposep> Ax=0has more variables than equations= Ax=0has nontrivial solutions= columns ofAare linearly dependentJiwen He, university of HoustonMath 2331, Linear Algebra13 / Linear IndependenceDefinition Matrix Columns Special CasesSpecial Cases: ExamplesExamplesWith the least amount of work possible, decide which of thefollowing sets of vectors are linearly independent and give a reasonfor each 321 , 964 b.

8 Columns of 1 2 3 4 56 7 8 9 09 8 7 6 54 3 2 1 8 Jiwen He, university of HoustonMath 2331, Linear Algebra14 / Linear IndependenceDefinition Matrix Columns Special CasesSpecial Cases: Examples (cont.)Examples (cont.)c. 321 , 963 , 000 d. 8214 Jiwen He, university of HoustonMath 2331, Linear Algebra15 / Linear IndependenceDefinition Matrix Columns Special CasesCharacterization of Linearly Dependent SetsExampleConsider the set of vectors{v1,v2,v3,v4}inR3in the followingdiagram. Is the set linearly dependent? ExplainJiwen He, university of HoustonMath 2331, Linear Algebra16 / Linear IndependenceDefinition Matrix Columns Special CasesCharacterization of Linearly Dependent SetsTheoremAn indexed setS={v1,v2.}

9 ,vp}of two or more vectors islinearly dependent if and only if at least one of the vectors inSis alinear combination of the others. In fact, ifSis linearlydependent, andv16=0, then some vectorvj(j 2) is a linearcombination of the preceding vectorsv1,..,vj He, university of HoustonMath 2331, Linear Algebra17 / 17