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# MATH-M303 Section 1.7 Notes MATH-M 303

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This 2 page Class Notes was uploaded by Kathryn Brinser on Thursday September 29, 2016. The Class Notes belongs to MATH-M 303 at Indiana University taught by Keenan Kidwell in Summer 2016. Since its upload, it has received 4 views. For similar materials see Linear Algebra for Undergraduate in Mathematics at Indiana University.

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Date Created: 09/29/16

M303 Section 1.7 Notes- Linear Independence 9-14-16 Linear independence- set of vectors???????? ???????? ,…,???? linearly independent if vector equation ???? ???? + ???? ???? + ⋯+ ???? ???? = ???? has only trivial solution ???? = ????, ie. ???? ,???? ,…,???? = 0 1 ???? 2 ???? ???? ???? 1 2 ???? o If ???? = ???????????? …???????? , t????en ???????? = ???? has same solution as ???? ????1 1???? ???? +2 2 ???? ???? = ???????? ???? Columns of ???? linearly independent iff ???????? = ???? has only trivial solution and iff no free variables present o No vector in set is linear combination of the others o Property that non-empty set of vectors may or may not have o In space/plane, vectors truly point in different directions Linear dependence- vectors ????????,????????,…,???? ????inearly dependent if 1 ???? + ????2???? ???? ⋯+ ???? ???? ???? ???? has non- trivial solutions, ie. if ???? ,???? ,…,???? not all zero 1 2 ???? o ????1????????+ ???? 2 +????⋯+ ???? ???? =???????? ????here some scalar(s) not zero known as dependencerelation Practical Test for Independence: o Form matrix ???? = ???? ???? …???????? , re????uce to EF o If no free variables present, vectors linearly independent o Otherwise, linearly dependent If ???? dependent, any nontrivial solution to ???????? = ???? gives dependence relation o Can also interpret homogeneous ???????? = ???? as vector equation ???? ???? + ???? ???? + ⋯+ ???? ???? = ???? 1 ???? 2 ???? ???? ???? Ex. ????11,2,3 + ???? 425,6 + ???? 2,130 = 0,0,0 ( ) o Always have trivial solution ???? = 0,0,0 ; linear independence means this is the only solution Ex. Given the vectors???????? = 1,2,3 ,???? ???? 4,5,6 ,???? = ????,1,0 : ) o Determine if the set ???? ,???? ,????} is linearly independent ????1 ???? ???? ???? ][ 2 = ???? ???? ???? ???? ????3 1 4 2 0 ???? 0 −2 0 2 5 1 0 ] ~ REF [0 ???? 1 0 ] 3 6 0 0 0 0 0 0 Since system has nontrivial solutions (system consistent), vectors linearly dependent o If possible, find a dependence relation amo????g ???? ,???? , a????d ???? ????1= 2???? 3 Solution set:????2= −???? 3 infinite number of solutions ????3= ???????????????? Trivial solution does not give dependence relation Let 3 = 1 so that ???? = 2,−1,1 : 2 1,2,3 − 4,5,6 + 2,1,0 ) = 2,4,6 − 4,5,6 + 2,1,0 ) = −2,−1,0 + 2,1,0 ) ( ) = 0,0,0 Ex. Are the columns of the following matrix ???? linearly independent? 0 1 4 ???? 2 −1 o ???? = [1 2 −1 ]~ EF [0 ???? 4 ] 5 8 0 0 0 ???????? o No free variables- only trivial solution works, so columns linearly independent Set ???? with 1 vector linearly independent as long as ???? ≠ ???? o Dependence relation would be ???????? = ???? where ???? ???? ℝ Ex. Determine if the following sets of vectors are linearly independent: o ???? = 3,1 , ???? = 6,2( ) ???? ???? ????????= 2???? ???? ????????????− ???? ???? ???? Fact of being multiples immediately led to linear dependence o ???? ???? 3,2 , ???? = ????,2 ( ) Not multiples of each other Suppose we have a dependence relation ????1????????+ ???? 2 ???? ???? where at least one of ????1,????2≠ 0 Assume ???? ≠ 0 1 ????1????????= −???? ????2 ???? ???? = −????2???? contradiction; vectors are not multiples ???? ????1 ???? Cannot have dependence relation; vectors linearly independent From example above, for set ????????,????????, vectors linearly independent if???? ???? ≠ ???????? where ???? ???? ℝ For > 2 vectors in general, must use matrix row reduction Theorem- set ???? = ???? 1???? ????…,???? ???? }of at least 2 vectors linearly dependent iff at least 1 vector in ???? is a linear combination of others o If ???? linearly dependent and???????? ≠ ????, then ???? where ???? > 1 is a linear combination ????f ???? ,???? ,????−???? o Ex. The set of vectors(3,1,0 , −1,2,1 , −11,1,2 )} is linearly dependent. Express one of t????e ???? as a linear combination of the other 2 and show that the set is linearly dependent. Given dependence relation????1???? ???? ???? ????2+???????? ???? 3 ????, solve for 1 vector −????2 3 ????????= ???? ???? ???? ???? ???????? −????1 3 ????????= ???? ???? ???? ???? ???????? −????1 2 ????????= ???? ???? ???? ???? ???????? 3 3 ???? ???? 1 3 −1 −11 1 To show dependence, find nontrivial solution to???????? ???? ???? ????][ 2 = [1 2 1 ][ 2 = ???? ????3 0 1 2 ????3 (need free variable(s)) 3 −1 −11 0 1 0 −3 0 [1 2 1 |0] ~ REF [0 1 2 0 ] 0 1 2 0 0 0 0 0 ????1= 3???? 3 {????2= −2???? 3 ???? = ???????????????? 3 ???? = 3???? 3−2???? ,3 3)= ???? 3,−2,1 ) Special Cases of Automatic Linear Dependence ???? o Theorem- Any set ???? ,???? ,????,???? ????}of ???? vectors in ℝ where ???? > ???? is linearly dependent Representative ???? × ???? matrix would have more columns than rows; always has free variable(s) ???? … ???? 11 1???? [ ⋮ ⋱ ⋮ ] has at most ???? pivots, at least ???? − ???? columns lack pivots ???? … ???? ????1 ???????? o Theorem- If the set ????????,????????,…,???? ????} contains ????, then it is linearly dependent Say ???? = ????; then ???? ???? + 0???? + 0???? + ⋯+ 0???? = ???? is valid dependence relation with ???? 1 ???? ???? ???? ???? nontrivial solutions Ex. Why are the vectors{(1,2,1,0 , −1,1,0,1 } linearly independent? o If we have dependence relation 1 1 + ????2 2= ????, what are the 3 and 4 entries of ????????? 1 −1 ????1− ???? 2 ???? 2???? + ???? o [ 2 1 ][ ] = [ 1 2] 1 0 ????2 ????1+ 0???? 2 0 1 0???? + ???? rd 1 2 In 3 entry, 2 = 0 In 4 entry, 1 = 0 Because ????1,????2= 0, system only has trivial solution o Notice that ????????has 1 in spot where ???? has 0

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