Math 550 Notes. Chapter 2. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Transcription

1 Math 550 Notes Chapter 2 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 2 Fall / 20

2 Linear algebra deals with finite dimensional vector spaces. Reminder F denotes R or C. V is a vector space over F. (Tarleton State University) Math 550 Chapter 2 Fall / 20

3 Outline 1 Span and Linear Independence 2 Bases 3 Dimension (Tarleton State University) Math 550 Chapter 2 Fall / 20

4 Linear Combinations and Span Definition Let (v 1,..., v m ) be a list of vectors in V. A linear combination of (v 1,..., v m ) is a vector of the form where a 1,..., a m F. a 1 v a m v m, The set of all such linear combinations is span(v 1,..., v m ). span(v 1,..., v m ) = {a 1 v a m v m a 1,..., a m F} span(v 1,..., v m ) is the smallest subspace of V containing v 1,..., v m. Special case: span() = {0}. (Tarleton State University) Math 550 Chapter 2 Fall / 20

5 Finite Dimensional Spaces Definition If span(v 1,..., v m ) = V, we say (v 1,..., v m ) spans V and that V is finite dimensional. Example In F n, define e j to be the vector whose jth coordinate is 1 and whose other coordinates are 0. e 1 = (1, 0,..., 0) e 2 = (0, 1,..., 0). e n = (0, 0,..., 1) Then F n = span(e 1,..., e n ), so F n is finite dimensional. (e 1,..., e n ) is called the standard basis for F n. (Tarleton State University) Math 550 Chapter 2 Fall / 20

6 Example Let P m (F) be the set of all polynomials with coefficients in F and degree at most m. Then the polynomials 1, z,..., z m span P m (F), so P m (F) is finite dimensional. P(F) is infinite dimensional. F is infinite dimensional. (Tarleton State University) Math 550 Chapter 2 Fall / 20

7 Linear Independence Definition Let v 1,..., v m V. (v 1,..., v m ) is linearly independent if, for any a 1,..., a m F, a 1 v a m v m = 0 implies a 1 = = a m = 0. Otherwise, we say (v 1,..., v m ) is linearly dependent. (v 1,..., v m ) is linearly dependent if there exist a 1,..., a m F such that a 1 v a m v m = 0 and a j 0, for some j {1,..., m}. Any list with 0 is linearly dependent. (v 1 ) is linearly independent iff v 1 0. (v 1, v 2 ) is linearly independent iff neither vector is a scalar multiple of the other. (Tarleton State University) Math 550 Chapter 2 Fall / 20

8 If vectors are removed from a list of linearly independent vectors, the list remains linearly independent. () is linearly independent. (Tarleton State University) Math 550 Chapter 2 Fall / 20

9 Linear Dependence Lemma and a Key Result Lemma Suppose (v 1,..., v m ) is linearly dependent in V, and v 1 0. Then there exists j {2,..., m} such that the following hold: v j span(v 1,..., v j 1 ); if the jth term is removed from (v1,..., v m ), the span of the remaining list equals span(v 1,..., v m ). Theorem Suppose V is finite dimensional, (u 1,..., u m ) is linearly independent in V, and (w 1,..., w n ) spans V. Then m n. That is, linearly independent lists are never longer than spanning lists. (Tarleton State University) Math 550 Chapter 2 Fall / 20

10 Proposition Every subspace of a finite dimensional vector space is finite dimensional. (Tarleton State University) Math 550 Chapter 2 Fall / 20

11 Outline 1 Span and Linear Independence 2 Bases 3 Dimension (Tarleton State University) Math 550 Chapter 2 Fall / 20

12 Bases Definition A list of vectors (v 1,..., v n ) in V is a basis of V if (v 1,..., v n ) is linearly independent, and (v 1,..., v n ) spans V. Example The standard basis (e 1,..., e n ) is a basis of F n. ((1, 2), (3, 5)) is a basis of F 2. ((1, 2)) is not a basis of F 2, since it doesn t span F 2. ((1, 2), (3, 5), (4, 7)) is not a basis of F 2, because it isn t linearly independent. (1, z,..., z m ) is a basis of P m (F). (Tarleton State University) Math 550 Chapter 2 Fall / 20

13 Proposition A list of vectors (v 1,..., v n ) in V is a basis iff every v V can be written uniquely in the form v = a 1 v a n v n, where a 1,..., a n F. (Tarleton State University) Math 550 Chapter 2 Fall / 20

14 Reducing Spanning Lists and Extending Linearly Independent Lists Proposition Every spanning list in V can be reduced to a basis of V. Corollary Every finite-dimensional vector space has a basis. Proposition Every linearly independent list of vectors in a finite dimensional vector space can be extended to a basis of the vector space. (Tarleton State University) Math 550 Chapter 2 Fall / 20

15 Proposition Suppose V is a finite dimensional vector space, and U is a subspace of V. Then there is a subspace W of V such that V = U W. (Tarleton State University) Math 550 Chapter 2 Fall / 20

16 Outline 1 Span and Linear Independence 2 Bases 3 Dimension (Tarleton State University) Math 550 Chapter 2 Fall / 20

17 Dimension Theorem Any two bases of a finite dimensional vector space have the same length. Definition The dimension of a finite dimensional vector space V is the length of any basis of V. dim V denotes the dimension of V. Example dim F n = n. dim P m (F) = m + 1. (Tarleton State University) Math 550 Chapter 2 Fall / 20

18 Proposition If V is finite dimensional, and U is a subspace of V, then dim U dim V. Proposition Suppose dim V = n. Then any spanning list of length n is a basis of V. Also, any linearly independent list of length n is a basis of V. Example ((5, 7), (4, 3)) is a basis for F 2. (Tarleton State University) Math 550 Chapter 2 Fall / 20

19 Two Equations Involving Dimension Theorem If U 1 and U 2 are subspaces of a finite dimensional vector space, then dim(u 1 + U 2 ) = dim U 1 + dim U 2 dim(u 1 U 2 ). Proposition Suppose V is finite dimensional, and V = U U m, where each U j is a subspace of V. Then dim V = dim U dim U m iff V = U 1 U m. Note: this proposition combines Proposition 2.19 and Exercise 17. (Tarleton State University) Math 550 Chapter 2 Fall / 20

20 Hint for Problem 6 on p. 35. For every j = 1, 2,..., define f j (x) = x j. Then show that for any m = 1, 2,..., the functions (f 1,..., f m ) are linearly independent. Hint: use Corollary 4.3 on p. 65. (Tarleton State University) Math 550 Chapter 2 Fall / 20