Linear Independence and Linear Dependence

Transcription

1 These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working through the exercises therein. Linear Independence and Linear Dependence Definition An indexed set of vectors v 1,v 2,,v n in m is said to be linearly independent if the vector equation x 1 v 1 x 2 v 2 x n v n m has only the trivial solution (x 1 x 2 x n ). If the above vector equation has non trivial solutions, then the set of vectors v 1,v 2,,v n is said to be linearly dependent and any equation of the form c 1 v 1 c 2 v 2 c n v n m with not all of the numbers c 1,c 2,,c n equal to zero is called a linear dependence relation for the set v 1,v 2,,v n. 1

2 Example Let v 1, v 2 5, and v Determine whether the set v 1,v 2,v 3 is linearly independent or linearly dependent. 2. If the set v 1,v 2,v 3 is linearly dependent, then write a linear dependence relation for this set. 2

3 Example Let v 1, v 2 3, and v Determine whether the set v 1,v 2,v 3 is linearly independent or linearly dependent. 2. If the set v 1,v 2,v 3 is linearly dependent, then write a linear dependence relation for this set. 3

4 Remark Suppose that A is an m n matrix. Then the set of vectors (in m ) that make up the columns of A form a linearly independent set if and only if the homogeneous matrix equation Ax m has only the trivial solution.

5 Determining Linear Independence/Dependence For Sets of One or Two Vectors One Vector Suppose that we have a single vector v 1 m. We can easily tell whether the set v 1 is linearly independent or linearly dependent: If v 1 m, then we have the linear dependence relation 1 v 1 m which shows that the set v 1 is linearly dependent. However, if v 1 m, then the vector equation x 1 v 1 m has only the trivial solution, which means that the set v 1 is linearly independent. Summary 1. If v 1 m, then the set v 1 is linearly dependent. 2. If v 1 m, then the set v 1 is linearly independent. Two Vectors Suppose that we have two vectors v 1 and v 2 m. We can easily tell whether the set v 1,v 2 is linearly independent or linearly dependent: Suppose that v 2 is a scalar multiple of v 1. Then v 2 cv 1 for some scalar c. This means that we have the linear dependence relation c v 1 1 v 2 m and hence that the set v 1,v 2 is linearly dependent. Likewise, if v 1 is a scalar multiple of v 2, then the set v 1,v 2 is linearly dependent. Conversely, suppose that the set v 1,v 2 is linearly dependent. Then we can write some linear dependence relation c 1 v 1 c 2 v 2 m where either c 1 or c 2. Ifc 1, then v 2 c 2 c 1 v 1 which means that v 2 is a scalar multiple of v 1.Ifc 2, then v 1 c 1 c 2 v 2 which means that v 1 is a scalar multiple of v 2. Thus if the set v 1,v 2 is linearly dependent, then either v 2 must be a scalar multiple of v 1 or v 1 must be a scalar multiple of v 2. Summary 1. If v 2 is a scalar multiple of v 1 or if v 1 is a scalar multiple of v 2, then the set v 1,v 2 is linearly dependent. 5

6 2. If neither of the vectors v 1 and v 2 is a scalar multiple of the other one, then the set v 1,v 2 is linearly independent. 6

7 Example Let 1 2 v 1 and v 2. 8 Is the set of vectors v 1,v 2 a linearly independent set or a linearly dependent set? If it is linearly dependent, then write a linear dependence relation for this set. 7

8 Determining Linear Independence/Dependence For Sets Containing the Zero Vector If we have a set of vectors v 1,v 2,,v n in m and one of the vectors in this set, say v j, is the zero vector ( m ), then this set is linearly dependent because we have the linear dependence relation v 1 v 2 1 v j v n m. Example Suppose that 1 3 v 1 6 1, v 2 6 1, v 3, and v Write a linear dependence relation for the set of vectors v 1,v 2,v 3,v. 8

9 Determining Linear Independence/Dependence For Sets Containing More Vectors Than There Are Entries in Each Vector If we have a set of vectors v 1,v 2,,v n in m and n m (in other words, there are more vectors in this set than there are entries in each vector, then the set v 1,v 2,,v n is linearly dependent. Here is why: If we let A be the m n matrix whose columns are the vectors v 1, v 2,,v n, then A has m rows and hence can have at most m pivot positions. Since A has n columns and n m, then not every column of A can have a pivot position. This means that the homogeneous matrix equation Ax m has non trivial solutions, and hence that the set of vectors v 1,v 2,,v n is linearly dependent. 9

10 Example Let v 1, v 2, and v Explain (without doing any computations) how you know that the set of vectors v 1,v 2,v 3 is linearly dependent. Then write a linear dependence relation for this set. (This latter part does require computation.) 1

11 A General Characterization of Linear Dependence Theorem Suppose that v 1,v 2,,v n is a set of two or more vectors in m. This set of vectors is linearly dependent if and only if at least one of the vectors in this set is a linear combination of the other vectors in the set. Furthermore, if the set v 1,v 2,,v n is linearly dependent and v 1 m, then there is a vector v j in this set (for some j 1) such that v j is a linear combination of the preceding vectors (v 1,v 2,,v j 1 ). Proof Suppose that v 1,v 2,,v n is a set of two or more vectors in m and suppose that this set of vectors is linearly dependent. Then we have a linear dependence relation c 1 v 1 c 2 v 2 c n v n m. Not all of the numbers c 1, c 2,, c n are zero. In particular, there is some index j such that. This means that v j c 1 v 1 c 2 v 2 1 v j 1 1 v j 1 c n showing that v j is a linear combination of the other vectors in the set. Conversely, suppose that there is some index j such that v j is a linear combination of the other vectors in the set. Then v j c 1 v 2 c 2 v 2 1 v j 1 1 v j 1 c n v n This means that c 1 v 2 c 2 v 2 1 v j 1 1 v j 1 v j 1 c n v n m and hence that the set v 1,v 2,,v n is linearly dependent. This completes the proof of the first part of the theorem. To prove the second statement of the theorem, suppose that the set v 1,v 2,,v n is linearly dependent and that v 1 m. Then there is a linear dependence relation c 1 v 1 c 2 v 2 c n v n m where, of course, not all of the numbers c 1,c 2,,c n are zero. Since v 1 m,it must in fact be true that not all of the numbers c 2,,c n are zero. Let j be the largest index in the set 2,,n such that. Then c 1 v 1 c 2 v 2 v j m and since, we have v j c 1 v 1 c 2 v 2 1 v j 1, showing that v j is a linear combination of the vectors preceding it. v n 11

12 Example Let v 1 1, v 2 1 2, v 3 1, v Explain why the set of vectors v 1,v 2,v 3,v is linearly dependent and find an index j 1,2,3, such that v j is a linear combination of the vectors preceding it. 12

13 Some Words of Caution 1. We know that if a set of vectors contains more vectors than there are entries in each vector, then this set must be linearly dependent. However, there are certainly linearly dependent sets of vectors that do not contain more vectors than there are entries in each vector. An example of a set of vectors that is linearly dependent but does not contain more vectors than there are entries in each vectors is: 2. We know that if a set of vectors is linearly dependent, then it must be true that at least one vector in the set is a linear combination of the other vectors in the set. However, it need not be true that all of the vectors in the set are linear combinations of the other vectors in the set. An example of a set of vectors that is linearly dependent but which contains a vector that is not a linear combination of the other vectors is: 13