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Nov 18, 2022 •387 likes •571 views

Section 4.3 d i E Linear Independent Vectors a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 211: Linear Algebra Dr. Abdulla Eid (University of Bahrain) Linearly Independence 1 / 17 d Goal: i E 1 Define Linearly

  1. Section 4.3 d i E Linear Independent Vectors a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 211: Linear Algebra Dr. Abdulla Eid (University of Bahrain) Linearly Independence 1 / 17

  2. d Goal: i E 1 Define Linearly independent and linearly dependent. a l 2 From dependent to independent. l u d 3 Independent in Maps ( R , R ) b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 2 / 17

  3. Subspace Definition 1 Let V be a vector space. v 1 , v 2 , . . . , v n are called linearly independent vectors if the equation d i k 1 v 1 + k 2 v 2 + . . . | k n v n = 0 E a has only the unique solution k 1 = 0, k 2 = 0, . . . , k n = 0 (called the trivial l l u solution ). d b A Note: This means k 1 , k 2 , . . . , k n are forced to be zero. . Definition 2 r D Let V be a vector space. v 1 , v 2 , . . . , v n are called linearly dependent vectors if the equation k 1 v 1 + k 2 v 2 + . . . k n v n = 0 has other solution than k 1 = 0, k 2 = 0, . . . , k n = 0 (called the nontrivial solution ). Dr. Abdulla Eid (University of Bahrain) Linearly Independence 3 / 17

  4. Example 3       1 0 0  are Determine whether the vectors e 1 = 0  , e 2 = 1  , e 3 = 0    0 0 1 d linearly independent in R 3 or not. i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 4 / 17

  5. Example 4       1 5 3 Determine whether the vectors v 1 = − 2  , v 2 = 6  , v 3 = 2     − 1 3 1 d are linearly independent in R 3 or not. i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 5 / 17

  6. Example 5  1   4   5  2 9 8       Determine whether the vectors v 1 =  , v 2 =  , v 3 =       2 9 9     d − 1 − 4 − 5 i are linearly independent in R 4 or not. E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 6 / 17

  7. Example 6 Determine whether the vectors P 1 = 1, P 2 = X , P 3 = X 2 , . . . , P n = X n are linearly independent in P n or not. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 7 / 17

  8. Example 7 Determine whether the vectors P 1 = 1 − X , P 2 = 5 + 3 X − 2 X 2 , P 3 = 1 + 3 X − X 2 are linearly independent in P 2 or not. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 8 / 17

  9. From Dependent to Independent Theorem 8 The set { v 1 , v 2 , . . . , v n } is linearly independent if and only if at least one d of the vector is expressible as linear combination of the rest. i E a l l u d b A . Corollary 9 r D Let { v 1 , v 2 , . . . , v n } be a linearly dependent set with v 1 = k 2 v 2 + · · · + k n v n , then span { v 1 , v 2 , . . . , v n } = span { v 2 , v 3 , . . . , v n } Dr. Abdulla Eid (University of Bahrain) Linearly Independence 9 / 17

  10. Example 10       1 2 1 Determine whether the vectors v 1 = 2  , v 2 = 4  , v 3 = 0  ,    3 6 1   1 d  are linearly independent in R 3 or not. If not, find an v 4 = 4 i E  5 a independent set from these vectors that gives the same span. l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 10 / 17

  11. Theorem 11 d i The set { v 1 , v 2 , . . . , v r } of vectors in R n with r > n is linearly dependent. E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 11 / 17

  12. Theorem 12 1 A set containing 0 is linearly dependent. d 2 A set with exactly one vector is linearly independent if and only if i E that vector is not 0 . a l 3 A set with exactly two vectors if and only if neither vector is a scalar l u d multiple of the other. b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 12 / 17

  13. Independent in Maps ( R , R ) Definition 13 Let f 1 , f 2 , . . . , f n are functions that are ( n − 1 ) differentiable functions. d i The determinant E a f 1 ( x ) f 2 ( x ) f n ( x )   l . . . l f ′ u f ′ f ′ 1 ( x ) 2 ( x ) n ( x ) . . .  d  f ′′ f ′′ f ′′  1 ( x ) 2 ( x ) n ( x )  b . . .   A   W f 1 , f 2 , ... , f n ( x ) : = det · · · . . .     · · · . . . .   r D   · · · . . .   f n − 1 f n − 1 f n − 1 ( x ) ( x ) ( x ) . . . 1 2 n is called the Wronskian of f 1 , f 2 , . . . , f n . Dr. Abdulla Eid (University of Bahrain) Linearly Independence 13 / 17

  14. d i E Theorem 14 a If f 1 , f 2 , . . . , f n have n − 1 continuous derivatives with a nonzero l l u Wronskian, then these functions are linearly independent. d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 14 / 17

  15. Example 15 Determine whether the vectors f 1 = 6, f 2 = 4 sin 2 x , f 3 = 3 cos 2 x are linearly independent in Maps ( R , R ) or not. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 15 / 17

  16. Example 16 Determine whether the vectors f 1 = x , f 2 = e x , f 3 = e − x are linearly independent in Maps ( R , R ) or not. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 16 / 17

  17. d i E a l l u Do HOMEWORK 1 d b A . r D Dr. Abdulla Eid (University of Bahrain) Linearly Independence 17 / 17

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