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Class Note for MATH 3321 at UH 2

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Date Created: 02/06/15
57 Vectors Linear Dependence and Linear Independence In Section 54 vectors were de ned either as l X n matrices which we called row vectors or as m X 1 matrices which we called column vectors In this section we want to think of vectors as simply being ordered ntuples of real numbers Most of the time will write vectors as a row vectors but occasionally it will be convenient to write them as columnsl Our main interest is in the concept of linear dependencelinear independence of sets of vectors At the end of the section we will extend this concept to linear dependenceindependence of sets of functions We denote the set of ordered n tuples of real numbers by the symbol Rnl That is R a1 a2 a3 i Wan l a1 a2 a3 Hi an are real numbers In particular R2 a b l a 12 real numbers which we can identify with the set of all points in the plane and R3 a b c la b c real numbers which we can identify with the set of points in space We will use lower case boldface letters to denote vectors The entries of a vector are called the components of the vector The operations of addition and multiplication by a number scalar that we de ned for matrices in Section 54 hold automatically for vectors since a vector is a matrix Addition is de ned only for vectors with the same number of components For any two vectors u a1 a2 a3 i i i an and V b1 b2 b3 i lbn in R we have 11Va17 a27 a37 7anb17 1127 b37 7b a1b17 a2b27 a3b37u 7 anbn and for any real number A AV Aa1 a2 a3l an Aa1a2 Aa3luanl Clearly the sum of two vectors in R is another vector in R and a scalar multiple of a vector in R is a vector in Rnl A sum of the form c1V1 c2V2 ckvk where V1 V2 H i Vk are vectors in R and c1 c2 i i i ck are real numbers is called a linear combination of V1 V2 Hi Vkl The set R together with the operations of addition and multiplication by a number is called a vector space of dimension n The term dimension n77 will become clear as we go on The zero vector in R which we7ll denote by 0 is the vector 0 0 0 0p 0 For any vector V E R we have V 0 0 V V the zero vector is the additive identity in Rnl Linear Dependence and Linear Independence In Chapter 3 we said that two functions f and g are linearly dependent if one of the functions is a scalar multiple of the other f and g are linearly independent if neither is a scalar multiple of the other Similarly let u and V be vectors in Rnl Then u and V are linearly dependent 214 if one of the vectors is a scalar multiple of the other eg u AV for some number A they are linearly independent if neither is a scalar multiple of the other Suppose that u V E R are linearly dependent With u AV Then uV implies uiAV0 This leads to an equivalent de nition of linear dependence u and V are linearly dependent if there exist two numbers cl and 52 not both zero such that 0111 62V 0 Note that if 0111 62V 0 and cl 0 then u 6261V AM This is the idea that we7ll use to de ne linear dependenceindependence in general DEFINITION 1 The set of vectors V1 V2 Vk in R is linearly dependent if there exist k numbers Cl 52 ck not all zero such that CiVi C2V2 Cka 0 0 is a nontrivial linear combination of V1 V2 Vk The set of vectors is linearly independent if it is not linearly dependent NOTE If there exists one set of k numbers Cl 52 ck not all zero then there exist in nitely many such sets For example 201 202 2 is another such set and cl C2 ck is another such set and so on The de nition of linear dependence can also be stated as The vectors V1 V2 Vk are linearly dependent if one of the vectors can be Written as a linear combination of the others For example if 61V1C2V2 Cka 0 and cl 0 then 52 CS Ck V1 V2 7V3quot39 Vk A2V23V3ka cl cl cl This form of the de nition parallels the de nition of the linear dependence of two vectors Stated in terms of linear independence the de nition can be stated equivalently as The set of vectors V1 V2 Vk is linearly independent if 61V1CQV2quot39Cka 0 implies cl 52 ck 0 ie the only linear combination of V1 V2 Vk Which equals 0 is the trivial linear combination The set of vectors is linearly dependent if it is not linearly independent 215 Linearly dependentindependent sets in R Example 1 Vectors in R a The vectors u 3 4 and V l 3 are linearly independent because neither is a multiple of the other b The vectors u 2 73 and V 76 9 are linearly dependent because V 73 u c Determine whether the vectors V1 l 2 V2 71 3 V3 5 7 are linearly dependent or linearly independent SOLUTION In this case we need to determine whether or not there are three numbers 01 02 03 not all zero such that 01V1 7 02V 7 03V3 0i Equating the components of the vector on the left and the vector on the right we get the homoge neous system of equations 01 7 02 503 0 201 302 703 0 We know that this system has nontrivial solutions since it is a homogeneous system with more unknowns than equations see Section 54 Thus the vectors must be linearly dependent The three parts of Example 1 hold in general any set of three or more vectors in R2 is linearly dependentl A set of two vectors in R2 is independent if neither vector is a multiple of the other dependent if one vector is a multiple of the other Example 2 Vectors in Rgi a The vectors u 3 4 72 and V 2 76 7 are linearly independent because neither is a multiple of the other b The vectors u 74 6 72 and V 2 73 l are linearly dependent because V 72 u c Determine whether the vectors V1 l 72 1 V2 2 l 71 V3 7 74 l are linearly dependent or linearly independent SOLUTION Method 1 We need to determine whether or not there are three numbers 01 02 03 not all zero such that 01V1 7 02V 7 03V3 0i Equating the components of the vector on the left and the vector on the right we get the homoge neous system of equations 01 202 703 0 7201 02 7 403 0 A 01 7 02 03 0 216 Writing the augmented matrix and reducing to row echelon form we get 1 2 7 0 1 2 7 0 72 1 74 0 A 0 1 2 0 1 71 1 0 0 0 0 0 The row echelon form implies that system A has in nitely nontrivial solutions Thus we can nd three numbers Cl 52 53 not all zero such that clvl C2V2 Cng 01 In fact we can nd in nitely many such sets Cl 52 031 Note that the vectors V1 V2 V3 appear as the columns of the coef cient matrix in system Method 2 Form the matrix with rows V1 V2 V3 and reduce to echelon form 1 72 1 1 72 1 2 1 71 a 0 1 735 7 74 1 0 0 0 The row of zeros indicates that the zero vector is a nontrivial linear combination of V1 V2 V3 Thus the vectors are linearly dependentl Method 3 Calculate the determinant of the matrix of coef cients 1 2 7 72 1 74 01 1 71 1 Therefore as we saw in the preceding section system A has in nitely many nontrivial solutionsl d Determine whether the vectors V1 1 2 73 V2 1 73 2 V3 2 71 5 are linearly dependent or linearly independent SOLUTION In part c we illustrated three methods for determining whether or not a set of vectors is linearly dependent or linearly independent We could use any one of the three methods here The determinant method is probably the easiest Since a determinant can be evaluated by expanding across any row or down any column it does not make any difference whether we write the vectors as rows or columns We7ll write them as rows 1 2 73 73 2 7301 71 Since the determinant is nonzero the only solution to the vector equation clvl C2V2 Cng 01 is the trivial solution the vectors are linearly independent Example 3 Determinewhether thevectors V1 1 2 74 V2 2 0 5 V3 1 71 7 V4 2 72 76 are linearly dependent or linearly independent 217 SOLUTION In this case we need to determine whether or not there are four numbers Cl 52 C3 54 not all zero such that CiVi C2V2 CsVs C4V4 0 Equating the components of the vector on the left and the vector on the right we get the homoge neous system of equations 012C2C32C4 0 201 7 C3 7 204 0 7451 502 703 7 664 0 We know that this system has nontrivial solutions since it is a homogeneous system with more unknowns than equationsi Thus the vectors must be linearly dependent Example 4 Vectors in R4 Let v1 2 0 71 4 v2 2 71 0 2 v3 72 4 73 4 v4 1 71 3 0 v5 0 1 75 3 a Determine whether the vectors V1 V2 V3 V4 V5 are linearly dependent of linearly indepen dent SOLUTION The vectors are linearly dependent because the vector equation CiVi C2V2 CsVs C4V4 65V5 0 leads to a homogeneous system with more unknowns than equationsi b Determine whether the vectors V1 V2 V3 V4 are linearly dependent or linearly independent SOLUTION To test for dependenceindependence in this case we have three options 1 Solve the system of equations CiVi C2V2 CsVs C4V4 0 A nontrivial solution implies that the vectors are linearly dependent if the trivial solution is the only solution then the vectors are linearly independent 2 Form the matrix A having V1 V2 V3 V4 as the rows and reduce to rowechelon formi If the rowechelon form has one or more rows of zeros the vectors are linearly dependent four nonzero rows means the vectors are linearly independent 3 Calculate det A detA 0 implies that the vectors are linearly dependent detA 0 implies that the vectors are linearly independent Options 1 and 2 are essentially equivalent the difference being that in option 1 the vectors appear as columnsi Option 2 requires a little less writing so we7ll use it 2 0 7l 4 l 7l 3 0 A 2 71 0 2 7 0 1 76 2 Verify this 72 4 73 4 0 0 l 0 l 7l 3 0 0 0 0 0 218 Therefore the vectors are linearly dependentl You can also check that det A 0 c Determine Whether V1 V2 V3 are linearly dependent or linearly independent SOLUTION Calculating a determinant is not an option here three vectors With four components do not form a square matrix We7ll row reduce 2 0 71 4 A 2 71 0 72 4 73 4 As you can verify 2 0 71 4 2 0 71 4 2 71 0 2 A 0 71 1 72 72 4 73 4 0 0 0 0 Therefore the vectors are linearly dependentl d Determine Whether the vectors V1 V2 V4 V5 are linearly dependent or linearly independent SOLUTION You can verify that 2 0 71 4 2 71 0 2 75 1 71 3 0 0 1 75 3 Therefore the vectors are linearly independent I In general suppose that V1 V2 1 i i Vk is a set of vectors in R 1 If k gt n the vectors are linearly dependentl 2 If k n Write the matrix A having V1 V2 Hi Vk as rowsl Either reduce A to row echelon form or calculate det A A row of zeros or det A 0 implies that the vectors are linearly dependent all rows nonzero or detA 0 implies that the vectors are linearly indep endentl 3 If k lt n Write the matrix A having V1 V2 Hi Vk as rows and reduce to row echelon forml A row of zeros implies that the vectors are linearly dependent all rows nonzero implies that the vectors are linearly independent Another look at systems of linear equations Consider the system of linear equations 219 1111 1212 1313 ainrn 1 1 2111 2212 2313 a2n1n 1 2 3111 3212 3313 asnrn 1 3 am111am212am313amn1n bm Note that we can write this system as the vector equation an L112 am hi L121 L122 a2n b2 11 i 12 i 39 39 39 In i i 7 am am am bm which is 11V1I2V2InVn b where an L112 am hi L121 L122 a2n b2 V1 V2 HiVn andb am am am bm Written in this form the question of solving the system of equations can be interpreted as asking whether or not the vector b can be written as a linear combination of the vectors V1 V2 i Vni As we know b may be written uniquely as a linear combination of V1 V2 i Vn the system has a unique solution b may not be expressible as a linear combination of V1 V2 Hi Vn the system has no solution or it may be possible to represent b as a linear combination of V1 V2 i i Vn in in nitely many different ways the system has in nitely many solutions Linear Dependence and Linear Independence of Functions As we saw in Chapter 3 two functions f and g are linearly dependent if one is a multiple of the other otherwise they are linearly independent DEFINITION 2 Let f1 f2 f3 Hi fn be functions de ned on an interval I The functions are linearly dependent if there exist n real numbers Cl 52 i i i an not all zero such that 61131 02131 Csf31 39 39 39CnfnI E 0 that is CHEW C2f21 CSfSI 0 for all z E 1 Otherwise the functions are linearly independent Equivalently the functions f1 f2 f3 i i fn are linearly independent if 61131 C2f21 Csf31quot39cnfn1 E 0 220 onlywhen 0152Cn0 Example 5 Let f1z l f2z z f3z 12 on I 70000 Show that f1 f2 f3 are linearly independent SOLUTION Suppose that the functions are linearly dependentl Then there exist three numbers Cl 52 Cg not all zero such that 0110215312E0i B Method 1 Since B holds for all I well let I 0 Then we have cl CQ 0 Cg 0 0 which implies cl 0 Since cl 0 B becomes 021 0312 E 0 or z 62 031 E 0 Since I is not identically zero we must have Cg 031 E 0i Letting z 0 we have Cg 0 Finally cl Cg 0 implies Cg 0 This contradicts our assumption that f1 f2 f3 are linearly dependentl Thus the functions are linearly independent Method 2 Since B holds for all I well evaluate at z 0 z l I 2 This yields the system of equations cl 0 61 C2 Cs 0 cl 202 403 0 It is easy to verify that the only solution of this system is cl Cg Cg 0 Thus the functions are linearly independent Method 3 Our functions are differentiable so well differentiate B twice to get 511621CSI2 E 0 Cg 2531 E 0 203 E 0 From the last equation Cg 0 Substituting Cg 0 in the second equation gives Cg 0 Substituting Cg 0 and Cg 0 in the rst equation gives cl 0 Thus B holds only when cl Cg Cg 0 which implies that the functions are linearly independent Example 6 Let sin I cos I sin I E n z 6 70000 Are these functions linearly dependent or linearly independent SOLUTION By the addition formula for the sine function 7rsinzcoslTrEcoszsianrl Ssinzilcosz 1 5111175 5 5 2 2 Since f3 isalinear combination of f1 and f2 we can conclude that f1 f2 f3 are linearly dependent on 70000 221 A Test for Linear Independence of Functions Wronskian Our test for linear independence is an extension of Method 3 in Example 5 THEOREM 1 Suppose that the functions fl7 f2 f3 i i i 7 n are n 7 ltimes differentiable on an interval It If the functions are linearly dependent on I then fl1 f2 1 fur WI 1 771 E 0 on It fin 1gtltzgt f n lgtltzgt MW Proof Since the functions are linearly dependent on I there exist n numbers Cl 52 m an not all zero7 such that 61131 02151 39 39 39 Cnfnlt gt 3 0A Differentiating this equation n 7 1 times7 we get the system of equations Cif1102f21quot39Cnfnr 7 0 01fz52f xcn lz E 0 61f r was cum 7 o A c1f 1gtltzgt c2f n 1gtltzgt enf n lgtltzgt 0 Choose any point a E I and consider the system of n equations in n unknowns 21 22 i i i 29 f1a21 f2a22 fna2n 0 fla21 130022 f a n 0 a l f2a22 fJLa n 0 f 1gta21 f 1gta22 ffln 1azn 0 This is a homogeneous system which7 from A7 has a nontrivial solution Cl 52 i i i cni Therefore7 as we showed in Section 567 f1a f2 a i i i fn a fla Ma A A A 13201 01 2 A A A fJLa 0i f 1gtltagt f n 1a m f 1a Since a was my point on I we conclude that the determinant is zero for all points in It I Recall from Chapter 3 that the determinant f1 f2 fl f2 is called the Wronskian of fl7 fgi The same terminology is used here 222 DEFINITION 3 Suppose that the functions f1 f2 f3 I II fn are n 7 ltimes differentiable on an interval I I The determinant 131 1021 m fnI fl1 f2 1 m IcyXI finil f nil I I I f fznil1 is called the Wronskz39zm of f1 f2 f3 III Theorem 1 can be stated equivalently as COROLLARY Suppose that the functions f1 f2 f3 III fn are n7 ltimes differentiable on an interval I and let be their WronskianI If f 0 for at least one I E I then the functions are linearly independent on I I This is a useful test for determining the linear independence of a set of functions Example 7 Show that the functions E l z 12 f4z 13 are linearly independentI SOLUTION These functions are threetimes differentiable on 700 Their Wronskian is l I 12 13 2 0 l 21 31 12 0 0 2 6x 0 0 0 6 Since W 0 the functions are linearly independentI Note You can use the Wronskian to show that any set of distinct powers of z is a linearly independent setI Caution Theorem 1 says that if a set of sufficiently differentiable functions is linearly dependent on an interval 1 then their Wronskian is identically zero on I The theorem does not say that if the Wronskian of a set of functions is identically zero on some interval then the functions are linearly dependent on that interval Here is an example of a pair of functions which are linearly independent and whose Wronskian is identically zero Example 8 Let fx x2 and let 712 72ltzlt0 I g 12 0 zlt2 on 72 2I The only question is whether 9 is differentiable at OI You can verify that it is Thus we can form their Wronskian For I Z 0 For I lt 0 1 ix WW 2x 721 W 53 Thus Wz30 on 722 We can state that f and g are linearly independent because neither is a constant multiple of the other g on 0 2 f 9 on 720i Another way to see this is Suppose that f and g are linearly dependent Then there exist two numbers Cl 52 not both zero such that clfz 5291 E 0 on 72 2 If we evaluate this identity at z l and z 71 we get the pair of equations cl Cg 0 cl 7 Cg 0 The only solution of this pair of equations is cl Cg 0 Thus f and g are linearly independent Exercises 57 1 Show that any set of vectors in R that contains the zero vector is a linearly dependent set Hint Show that the set 0 V2 V3 1 i i Vk is linearly dependent 2 Show that if the set V1 V2 V3 1 i i Vk is linearly dependent then one of the vectors can be written as a linear combination of the others Determine whether the set of vectors is linearly dependent or linearly independent If it is linearly dependent express one of the vectors as a linear combination of the others 3 1 72 3 72 4 1 74 8 4i 1 2 5 1 721 2 1 511 71 3 0 2 3 1 71 2 72 6 6i 1 2 73 1 73 2 2 71 7i 1 72 1 21 71 7 74 811 0 2 72 2 1 0 1 2 71 0 9 0 0 0 1 4 72 0 2 2 71 01 11 0 10 1 71 3 0 2 3 1 71 2 72 6 11 For which values of b are the vectors 1 b 3 71 linearly independent 12 For which values of b are the vectors 3 b 6 1271 linearly independent 13 For which values of b are the vectors 2 7b 2b6 4b linearly dependent 14 For which values of b are the vectors 1 12 2b 2 l 4 linearly independent 224 15 16 17 18 19 20 21 22 23 24 25 26 1 2 71 3 Consider the matrix A 0 1 71 2 7 Which is rowechelon form Show that the 0 0 0 1 row vectors of A are a linear independent set Are the nonzero row vectors of any matrix in rowechelon form linearly independent Let V1 and V2 be linearly independent Prove that V1V27 V17V2 are linearly independent Let S v17 vQ empty subset of S is also linearly independent Suppose that T V17 V27 1 i i vk be a linearly independent set of vectors Prove that every non vm is a linearly dependent set Is every nonempty subset of T linearly dependent Calculate the Wronskian of the set of functions Then determine Whether the functions are linearly dependent or linearly independent 1611 6 1621 Eb a 17 f1z sin a1 f2 cos am I 6 70000 a 0 131 17 131 127 fsr 063 131 17 131 1 17 fsr f2 f1z z 7 12 151 12 7 Sac f3z 2x 12 131 17 131 67 fsr re 131 617 131 6 fsr 62 121 egg 1 functions are linearly dependent on 1 True or false I 6 70000 1 6 70000 1 6 000 1 6 70000 1 6 70000 1 6 70000 1f the Wronskian of a set of functions is identically zero on an interval 1 then the b If a set of functions is linearly dependent on an interval 1 then the Wronskian of the functions is identically zero on 1 True or false c If the Wronskian of a set of functions is nonzero at some points of an interval I and zero at other points7 then the functions are linearly independent on 1 True or false Show that the functions f0z 17 z 12 indep endenti 7 fkr ask are linearly 225

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