1.7 Linear Independence McDonald Fall 2018, MATH 2210Q, 1.7Slides - PDF document

1.7 Linear Independence McDonald Fall 2018, MATH 2210Q, 1.7Slides 1.7 Homework : Read section and do the reading quiz. Start with practice problems, then do ❼ Hand in: 1, 5, 7, 15, 16, 20, 21 ❼ Extra Practice: 1-20 Definition 1.7.1. An indexed set of vectors S = { v 1 , . . . , v p } in R n is said to be linearly independent if the vector equation x 1 v 1 + x 2 v 2 + · · · + x p v p = 0 has only the trivial solution. S is linearly dependent if for some c 1 , . . . , c p not all zero c 1 v 1 + c 2 v 2 + · · · + c p v p = 0 .       1 2 0 Example 1.7.2. Let v 1 = 2  , v 2 = 1  , and v 3 = 3  .          0 0 3 Is { v 1 , v 2 , v 3 } linearly independent? If not, find a linear dependence relation.       1 4 2 Example 1.7.3. Let v 1 =  , v 2 =  , and v 3 =  . 2 5 1          3 6 0 Is { v 1 , v 2 , v 3 } linearly independent? If not, find a linear dependence relation. 1

� � Remark 1.7.4. If A = , then the homogeneous equation A x = 0 can be written v m · · · v m x 1 v 1 + · · · + x n v m = 0 . Thus, linear independence is the same as having no non-trivial solutions to this matrix equation. Definition 1.7.5. The columns of a matrix A are linearly independent if and only if A x = 0 has no non-trivial solutions.   0 1 4 Example 1.7.6. Determine if the columns of A = 1 2 − 1  are linearly independent.    5 8 0 Example 1.7.7. Determine if the following sets of vectors are linearly independent. � � � � � � � � 3 6 3 6 (a) v 1 = , v 2 = (b) v 1 = , v 2 = 1 2 2 2 Proposition 1.7.8 (Sets of two vectors) . A set of two vectors { v 1 , v 2 } is linearly inde- pendent if and only if neither of the vectors is a multiple of the other. 2

(Characterization of Linearly Dependent Sets) . An indexed set Theorem 1.7.9 { v 1 , · · · , v p } of two or more vectors is linearly dependent if and only if at least one of the vectors is a linear combination of the others. In fact, if S is linearly dependent and v 1 � = 0 , then some v j ( j > 1) is a linear combination of the preceding vectors, v 1 , . . . , v j − 1 . Example 1.7.10. If u and v are linearly independent non-zero vectors in R 3 . Geometrically describe Span { u , v } . Prove w is in Span { u , v } if and only if { u , v , w } is a linearly dependent set. 3

Theorem 1.7.11 (Too many vectors) . If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set { v 1 , . . . , v p } in R n is linearly dependent if p > n . Proof: Theorem 1.7.12. If a set S in R n contains the zero vector, then S is linearly dependent. Proof: Example 1.7.13. Determine by inspection (without matrices) if given sets are linearly dependent.               1 4 7 1 1 0 7 (a) 2  , 5  , 8  , 3 (b) 2  , 0  , 8                        3 6 9 5 3 0 9           1 2 7 1 2 (c) 2  , 4  , 8 (d) 2  , 4                  3 6 9 3 7 4