Lecture 13. Math 22 Summer 2017 July 17, 2017

Transcription

1 Lecture 13 Math 22 Summer 2017 July 17, 2017

2 Reminders/Announcements Midterm 1 Wednesday, July 19, Kemeny 008, 6pm-8pm Please respond to if you have a conflict with this time! Kate is moving her study group to Wednesday 3pm - 4:30pm in the usual place (Berry 370). She is not having study group this Sunday. Last scheduled x-hour meets tomorrow. Thursday office hours are on Tuesday this week due to: HW4 will be posted later today and due Friday as usual. Any other questions (content or otherwise) please feel free to me.

3 Just for today 4.1 Two examples we didn t get to last time 4.2 Null and column spaces corresponding to linear maps

4 4.1 Proving a set is a subspace

5 4.1 Proving a set is a subspace To show a subset of a vector space is a subspace we can always use the definition.

6 4.1 Proving a set is a subspace To show a subset of a vector space is a subspace we can always use the definition. However, the previous theorem gives us another way...

7 4.1 Proving a set is a subspace To show a subset of a vector space is a subspace we can always use the definition. However, the previous theorem gives us another way... Let s + 2t H = t : s, t R 3s 7t R3.

8 4.1 Proving a set is a subspace To show a subset of a vector space is a subspace we can always use the definition. However, the previous theorem gives us another way... Let s + 2t H = t : s, t R 3s 7t R3. How can we use the previous theorem to show H is a subspace?

9 4.1 Proving a set is a subspace To show a subset of a vector space is a subspace we can always use the definition. However, the previous theorem gives us another way... Let s + 2t H = t : s, t R 3s 7t R3. How can we use the previous theorem to show H is a subspace? Well, 1 2 H = Span 0,

10 4.1 Proving a set is not a subspace

11 4.1 Proving a set is not a subspace To show that a subset is not a subspace, we just need to show that at least one of the axioms fails to be satisfied.

12 4.1 Proving a set is not a subspace To show that a subset is not a subspace, we just need to show that at least one of the axioms fails to be satisfied. Let H = {[ ] } 3s : s R s

13 4.1 Proving a set is not a subspace To show that a subset is not a subspace, we just need to show that at least one of the axioms fails to be satisfied. Let H = {[ ] } 3s : s R s How do we show H is not a subspace?

14 4.2 Definition of null space

15 4.2 Definition of null space Definition

16 4.2 Definition of null space Definition Let A be an m n matrix.

17 4.2 Definition of null space Definition Let A be an m n matrix. The null space of A is the set of solutions to the matrix equation Ax = 0.

18 4.2 Definition of null space Definition Let A be an m n matrix. The null space of A is the set of solutions to the matrix equation Ax = 0. We denote the null space of a matrix A by Nul A.

19 4.2 Theorem 2

20 4.2 Theorem 2 Theorem

21 4.2 Theorem 2 Theorem Let A be an m n matrix.

22 4.2 Theorem 2 Theorem Let A be an m n matrix. Then the null space of A is a subspace.

23 4.2 Theorem 2 Theorem Let A be an m n matrix. Then the null space of A is a subspace. Of what vector space?

24 4.2 Theorem 2 Theorem Let A be an m n matrix. Then the null space of A is a subspace. Of what vector space? R n.

25 4.2 Theorem 2 Theorem Let A be an m n matrix. Then the null space of A is a subspace. Of what vector space? R n. Proof.

26 4.2 Theorem 2 Theorem Let A be an m n matrix. Then the null space of A is a subspace. Of what vector space? R n. Proof. Show 0 Nul A.

27 4.2 Theorem 2 Theorem Let A be an m n matrix. Then the null space of A is a subspace. Of what vector space? R n. Proof. Show 0 Nul A. Show Nul A closed under addition.

28 4.2 Theorem 2 Theorem Let A be an m n matrix. Then the null space of A is a subspace. Of what vector space? R n. Proof. Show 0 Nul A. Show Nul A closed under addition. Show Nul A closed under scalar multiplication.

29 4.2 An explicit description for Nul A Let A =

30 4.2 An explicit description for Nul A Let A = Find vectors u, v, w such that Nul A = Span{u, v, w}.

31 4.2 An explicit description for Nul A Let A = Find vectors u, v, w such that Nul A = Span{u, v, w}. Solution:

32 4.2 An explicit description for Nul A Let A = Find vectors u, v, w such that Nul A = Span{u, v, w}. Solution: First note that the RREF of A is

33 4.2 An explicit description for Nul A Let A = Find vectors u, v, w such that Nul A = Span{u, v, w}. Solution: First note that the RREF of A is Now write out the parametric vector form of the solutions to Ax = 0.

34 4.2 An explicit description for Nul A Let A = Find vectors u, v, w such that Nul A = Span{u, v, w}. Solution: First note that the RREF of A is Now write out the parametric vector form of the solutions to Ax = 0. Can you see how this yields a spanning set?

35 4.2 An explicit description for Nul A Let A = Find vectors u, v, w such that Nul A = Span{u, v, w}. Solution: First note that the RREF of A is Now write out the parametric vector form of the solutions to Ax = 0. Can you see how this yields a spanning set? Is this set linearly independent?

36 4.2 Null spaces as kernels of linear maps

37 4.2 Null spaces as kernels of linear maps As you might expect, Nul A can also be phrased in the language of linear maps.

38 4.2 Null spaces as kernels of linear maps As you might expect, Nul A can also be phrased in the language of linear maps. Definition

39 4.2 Null spaces as kernels of linear maps As you might expect, Nul A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces.

40 4.2 Null spaces as kernels of linear maps As you might expect, Nul A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces. (What is this?)

41 4.2 Null spaces as kernels of linear maps As you might expect, Nul A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces. (What is this?) The kernel of T is the set of all vectors x V such that T (x) = 0.

42 4.2 Null spaces as kernels of linear maps As you might expect, Nul A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces. (What is this?) The kernel of T is the set of all vectors x V such that T (x) = 0. We denote this set by ker T.

43 4.2 Null spaces as kernels of linear maps As you might expect, Nul A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces. (What is this?) The kernel of T is the set of all vectors x V such that T (x) = 0. We denote this set by ker T. How does ker T relate to Nul A?

44 4.2 Definition of column space

45 4.2 Definition of column space Definition

46 4.2 Definition of column space Definition Let A be an m n matrix.

47 4.2 Definition of column space Definition Let A be an m n matrix. The column space of A is the span of the columns of A.

48 4.2 Definition of column space Definition Let A be an m n matrix. The column space of A is the span of the columns of A. We denote this space as Col A.

49 4.2 Definition of column space Definition Let A be an m n matrix. The column space of A is the span of the columns of A. We denote this space as Col A. Why is Col A a subspace?

50 4.2 Definition of column space Definition Let A be an m n matrix. The column space of A is the span of the columns of A. We denote this space as Col A. Why is Col A a subspace? What vector space is Col A a subspace of?

51 4.2 Definition of column space Definition Let A be an m n matrix. The column space of A is the span of the columns of A. We denote this space as Col A. Why is Col A a subspace? What vector space is Col A a subspace of? When is Col A = R m?

52 4.2 Column spaces as images of linear maps

53 4.2 Column spaces as images of linear maps As you might expect, Col A can also be phrased in the language of linear maps.

54 4.2 Column spaces as images of linear maps As you might expect, Col A can also be phrased in the language of linear maps. Definition

55 4.2 Column spaces as images of linear maps As you might expect, Col A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces.

56 4.2 Column spaces as images of linear maps As you might expect, Col A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces. The image or range of T is the set of all vectors b W such that there exists x V and T (x) = b.

57 4.2 Column spaces as images of linear maps As you might expect, Col A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces. The image or range of T is the set of all vectors b W such that there exists x V and T (x) = b. We denote this set by imgt.

58 4.2 Column spaces as images of linear maps As you might expect, Col A can also be phrased in the language of linear maps. Definition Let T : V W be a linear map of vector spaces. The image or range of T is the set of all vectors b W such that there exists x V and T (x) = b. We denote this set by imgt. How does imgt relate to Col A?

59 4.2 Classwork Let A =

60 4.2 Classwork Let A = Nul A R k for what k? Col A R k for what k?

61 4.2 Classwork Let A = Nul A R k for what k? Col A R k for what k? 2. Find a nonzero vector in Nul A. Find a nonzero vector in Col A.

62 4.2 Classwork Let A = Nul A R k for what k? Col A R k for what k? 2. Find a nonzero vector in Nul A. Find a nonzero vector in Col A. 3. Let u = 0, v = Is u Nul A? Is u Col A? Is v Nul A? Is v Col A?