Lecture 03. Math 22 Summer 2017 Section 2 June 26, 2017

Transcription

1 Lecture 03 Math 22 Summer 2017 Section 2 June 26, 2017

2 Just for today (10 minutes) Review row reduction algorithm (40 minutes) 1.3 (15 minutes) Classwork

3 Review row reduction algorithm

4 Review row reduction algorithm Use row reduction to put the following matrix is RREF.

5 Review row reduction algorithm Use row reduction to put the following matrix is RREF / /

6 Review row reduction algorithm Use row reduction to put the following matrix is RREF / / How many pivots does this matrix have?

7 Review row reduction algorithm Use row reduction to put the following matrix is RREF / / How many pivots does this matrix have? How many free variables does this matrix have?

8 Review row reduction algorithm Use row reduction to put the following matrix is RREF / / How many pivots does this matrix have? How many free variables does this matrix have? Suppose this is the augmented matrix of a linear system. What can you say about the solution set?

9 Review row reduction algorithm Use row reduction to put the following matrix is RREF / / How many pivots does this matrix have? How many free variables does this matrix have? Suppose this is the augmented matrix of a linear system. What can you say about the solution set? Suppose this is the coefficient matrix of a linear system. What can you say about the solution set?

10 1.3 Vectors in Rn

11 1.3 Vectors in R n Recall vectors in R 2, R 3, R n.

12 1.3 Vectors in R n Recall vectors in R 2, R 3, R n. It is convenient in linear algebra to write vectors as column vectors.

13 1.3 Vectors in R n Recall vectors in R 2, R 3, R n. It is convenient in linear algebra to write vectors as column vectors. That is, as n 1 matrices.

14 1.3 Vectors in R n Recall vectors in R 2, R 3, R n. It is convenient in linear algebra to write vectors as column vectors. That is, as n 1 matrices. Recall the algebraic properties of vectors.

15 1.3 Vectors in R n Recall vectors in R 2, R 3, R n. It is convenient in linear algebra to write vectors as column vectors. That is, as n 1 matrices. Recall the algebraic properties of vectors. Examples?

16 1.3 Vectors in R n Recall vectors in R 2, R 3, R n. It is convenient in linear algebra to write vectors as column vectors. That is, as n 1 matrices. Recall the algebraic properties of vectors. Examples? What is the difference between a vector and a scalar?

17 1.3 Vectors in R n Recall vectors in R 2, R 3, R n. It is convenient in linear algebra to write vectors as column vectors. That is, as n 1 matrices. Recall the algebraic properties of vectors. Examples? What is the difference between a vector and a scalar? What does it mean for two vectors to be equal?

18 1.3 Linear combinations

19 1.3 Linear combinations Definition Given v 1,..., v p R n and given scalars c 1,..., c p R, we define the linear combination of v 1,..., v p with the weights c 1,..., c p by c 1 v c p v p.

20 1.3 Linear combinations Definition Given v 1,..., v p R n and given scalars c 1,..., c p R, we define the linear combination of v 1,..., v p with the weights c 1,..., c p by c 1 v c p v p. How can we interpret a linear combination of vectors geometrically?

21 1.3 Linear combinations Definition Given v 1,..., v p R n and given scalars c 1,..., c p R, we define the linear combination of v 1,..., v p with the weights c 1,..., c p by c 1 v c p v p. How can we interpret a linear combination of vectors geometrically? Let a, b R and v 1 = [ ] 1 0 and v 2 = [ ] 0. 1

22 1.3 Linear combinations Definition Given v 1,..., v p R n and given scalars c 1,..., c p R, we define the linear combination of v 1,..., v p with the weights c 1,..., c p by c 1 v c p v p. How can we interpret a linear combination of vectors geometrically? Let a, b R and v 1 = [ ] 1 0 and v 2 = [ ] 0. 1 What is av 1 + bv 2?

23 1.3 Linear span

24 1.3 Linear span Definition Let v 1,..., v p R n.

25 1.3 Linear span Definition Let v 1,..., v p R n. We define the span of a set of vectors as: Span{v 1,..., v p } := {c 1 v c p v p : c 1,..., c p R}.

26 1.3 Linear span Definition Let v 1,..., v p R n. We define the span of a set of vectors as: Span{v 1,..., v p } := {c 1 v c p v p : c 1,..., c p R}. Informally, the span of a set of vectors is the set of all linear combinations of those vectors.

27 1.3 Linear span Definition Let v 1,..., v p R n. We define the span of a set of vectors as: Span{v 1,..., v p } := {c 1 v c p v p : c 1,..., c p R}. Informally, the span of a set of vectors is the set of all linear combinations of those vectors. Span {[ ] 1, 0 [ ] 0, 1 [ ]} 1 =? 1

28 1.3 Linear span Definition Let v 1,..., v p R n. We define the span of a set of vectors as: Span{v 1,..., v p } := {c 1 v c p v p : c 1,..., c p R}. Informally, the span of a set of vectors is the set of all linear combinations of those vectors. {[ ] [ ] [ ]} Span,, =? Span 0, 1, =?

29 1.3 Some properties of span

30 1.3 Some properties of span Let u, v be nonzero vectors in R n.

31 1.3 Some properties of span Let u, v be nonzero vectors in R n. u, v Span{u, v}

32 1.3 Some properties of span Let u, v be nonzero vectors in R n. u, v Span{u, v} Span{u, v, 0} = Span{u, v}

33 1.3 Some properties of span Let u, v be nonzero vectors in R n. u, v Span{u, v} Span{u, v, 0} = Span{u, v} u ± v, cu Span{u, v}

34 1.3 Some properties of span Let u, v be nonzero vectors in R n. u, v Span{u, v} Span{u, v, 0} = Span{u, v} u ± v, cu Span{u, v} S, T R n and S T implies Span{S} Span{T }

35 1.3 Linear systems as linear combinations

36 1.3 Linear systems as linear combinations Consider the vector equation x 1 a x n a n = b.

37 1.3 Linear systems as linear combinations Consider the vector equation x 1 a x n a n = b. Consider the linear system whose augmented matrix has columns a i and b which we abbreviate [ ] a 1 a 2 b.

38 1.3 Linear systems as linear combinations Consider the vector equation x 1 a x n a n = b. Consider the linear system whose augmented matrix has columns a i and b which we abbreviate [ ] a 1 a 2 b. Next time we will prove the fundamental result that... these objects both have the same solution set!

39 1.3 Linear systems as linear combinations Consider the vector equation x 1 a x n a n = b. Consider the linear system whose augmented matrix has columns a i and b which we abbreviate [ ] a 1 a 2 b. Next time we will prove the fundamental result that... these objects both have the same solution set! This means that a vector b R m can be expressed as a linear combination of the vectors [ a 1,..., a n ] if any only if the linear system corresponding to a 1 a n b is consistent.

40 1.3 Classwork

41 1.3 Classwork Suppose [ ] a 1 a 2 a 3 b = /

42 1.3 Classwork Suppose [ ] a 1 a 2 a 3 b = Do there exist scalars x 1, x 2, x 3 R such that x 1 a 1 + x 2 a 2 + x 3 a 3 = b? /

43 1.3 Classwork

44 1.3 Classwork Consider A = , and b =

45 1.3 Classwork Consider A = , and b = Let W be the linear span of the columns of A.

46 1.3 Classwork Consider A = , and b = Let W be the linear span of the columns of A. Is b W?

47 1.3 Classwork Consider A = , and b = Let W be the linear span of the columns of A. Is b W? 0 Is 8 W? 2