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Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver P

Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, Second Edition.

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Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver P

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  1. Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, Second Edition

  2. Friday, February 14, Section 3.1 • Page 106 Problems 24,26,44 • Main Idea: Review what we have been doing and use new words. • Key Words: Kernel, Image, Codomain, Span, < V1, V2, ..., Vr > • Goal: Learn uses and limitations of the notation for the span of a set of vectors.

  3. We know how to solve AX = B. We reduce • it to the Row Canonical Form and read off the • answers. • The answer is of the form • X = X0 + a1 X1 + a2 X2 + ... ar Xr

  4. There are three general possibilities for the outcome. (1) There are no solutions at all. This happens when RHS has a stair step one. (2) There is exactly one solution. This happens when RHS is the only column with no stair step one. (3) There are infinitely many solutions. This happens when RHS and at least one other column has no stair step one.

  5. Case (1) implies that there must be some B's such that AX = B is not possible. We then ask the related question, “For what B's does there exist a solution for AX = B?” The answer to the question is a set. We actually know of how to express the solution right now.

  6. From our knowledge of matrix multiplication, AX = B B is a linear combination of the columns of the matrix A.

  7. The SPAN of a set of vectors is the set of all linear combinations of the vectors in the set. Thus we say: AX = B B is in the span of the columns of A.

  8. In matrix theory often we often have to answer questions where there are an infinite number of possible answers. A convenient way of expressing all answers is to give a few such answers and then say that the complete set of answers is the span of the vectors listed.

  9. Notice that when we solved AX = B, we wrote the answer as X = X0 + a1 X1 + a2 X2 + ... + ar Xr We said that the set of all answers were of the form: X = X0 + S where S was some element in <X1, X2, ..., Xr>.

  10. The solutions can be expressed in many, many ways. We could also express them differently as X = Y0 + b1 Y1 + b2 Y2 + ... + bs Ys or X = Y0 + T where T is some element in <Y1, Y2, ..., Ys>. The answers are hard to compare. They can both be correct, but look quite different.

  11. We can say that: 1. r = s. (If not, one can be removed without changing the span.) 2. <X1,X2, ..., Xr > = <Y1, Y2, ..., Ys> The two spans are equal, not the individual vectors. 3. X0 and Y0 can be replaced by any solution. Because of 2, we are interested in knowing when two spans are the same.

  12. In less mathematical terms, look at the forest, not the trees. The forest is the span. The trees are the individual vectors.

  13. | 2 | | 2 | |-1 | | 3 | Example. < | 3 |, |-4 | > = < | 2 |, | -2 | > | 5 | |-2 | | 1 | | 1 | These two spans are the same plane. The normal to the plane is the red line. If we change the view point we can see that the spans are the same plane.

  14. We can see that the spans are the same by | a | seeing which vectors | b | are in the span. | c | | 2 | | 2 | | a | |-1 | | 3 | | a | x| 3 | + y | -4 | = | b | x | 2 | + y | -2 | = | b | | 5 | | -2 | | c | | 1 | | 1 | | c |

  15. | 2 2 a | | -1 3 a | | 3 -4 b | | 2 -2 b | | 5 -2 c | | 1 1 c |

  16. | 2 2 a | | -1 3 a | | 3 -4 b | | 2 -2 b | | 5 -2 c | | 1 1 c | | 1 1 a/2 | | 1 -3 -a | | 3 -4 b | | 2 -2 b | | 5 -2 c | | 1 1 c |

  17. | 1 1 a/2 | | 1 -3 -a | | 0 -7 -3a/2+b | | 0 4 2a+b | | 0 -7 -7a/2+c | | 0 4 a+c | | 1 1 a/2 | | 1 -3 -a | | 0 -7 -3a/2+b | | 0 4 2a+b | | 0 0 - a-b+c | | 0 0 -a-b+c |

  18. | 1 0 2 a/7 + b/7 | | 1 0 a/2 + 3b/4 | | 0 1 +3 a/14 - b/7 | | 0 1 a/2 + b/4 | | 0 0 -a – b + c | | 0 0 - a - b + c | | a | The vector | b | is in the span | c | -a-b+c = 0, Which simply says: "the third coordinate is the sum of the first two.“

  19. Notice that this is true for all the vectors given. We were interested in showing that the two spans were the same and we have accomplished that. We can go on and show how to get the | a | vector | b | in each case. |a+b|

  20. | 2 | | 2 | | a | (2a/7 + b/7) | 3 | + (3a/14 - b/7) | -4 | = | b | | 5 | | -2 | |a+b| |-1 | | 3 | | a | (a/2 + 3b/4)| 2 | +(a/2 + b/4)|-2 | = | b | | 1 | | 1 | |a+b|

  21. The IMAGE of a matrix A, is the span of the columns of A. B is in the image of A AX = B has a solution, B is not in the image of A AX = B is inconsistent. Proof: This is self evident because AX is a linear combination of the columns of A.

  22. There is a distinction between kernel and null space, but in math 307, we ignore it so, in this course, Kernel is the same thing as the Null Space. (It is the null space of a matrix A, but the kernel of the linear transformation whose matrix is A) The NULL SPACE of a matrix A is the set of all vectors X such that AX = 0. The null space is < X1,X2, ..., Xr > for the system of linear equations AX = 0. (* NOTICE X0 is missing *)

  23. Example: For the matrix | 0 1 | | 0 0 |. What is the Range? What is the Kernel?

  24. Range = | c | | 0 | Kernel = | c | | 0 | Both the kernel and the range are the X-axis.

  25. For the matrix | 1 2 3 | | 2 1 0 | | 3 3 3 | What is the range? What is the kernel?

  26. |1 2 3 | | 1 2 3 | | 1 2 3 | | 1 0 -1 | |2 1 0 | ~ | 0 -3 -6 | ~ | 0 1 2 | ~ | 0 1 2 | |3 3 3 | | 0 -3 -6 | | 0 0 0 | | 0 0 0 |

  27. | 1 | The kernel is the span of |-2 |. | 1 | | 1 | | 2 | The range is the span of | 2 | | 1 | | 3 | | 3 |

  28. The third column is unnecessary because it can be generated from the first two. How?

  29. The elements of the null space are dependence relations. Since | 1 2 3 | | 1 | | 0 | | 2 1 0 | |-2 | = | 0 | | 3 3 3 | | 1 | | 0 | We can view this using the columns as: C1 - 2 C2 + C3 = 0 so: C3 = -C1 + 2 C2. Thus C3 is not needed since C3 is in the span of the first two.

  30. Which columns should you use in your spanning set? A useful rule is: Use the columns which generate stair step ones. The stair step ones determine the columns for the spanning set of the column space.

  31. The stair step ones determine the columns for the spanning set of the column space. The non-stair step ones determine the columns for the spanning set of the null space.

  32. Pictorially speaking. X-------------------->AX ------- Rm Rn 0 range kernel

  33. Question. What is the Column space of | 4 7 2 4 | A = | 3 5 1 2 | | 1 2 1 2 |

  34. It will be a subspace of R^3. | 4 7 2 4 x| | 1 2 1 2 z| | 1 2 1 2 z| |1 0-3-6 2y-5z| | 3 5 1 2 y|~| 3 5 1 2 y|~| 0-1-2-4 y-3z|~|0 1 2 4 -y+3z| | 1 2 1 2 z| | 4 7 2 4 x| | 0-1-2-4 x-4z| |0 0 0 0 x-y-z| | x | | y+z| | y | is in the range if and only if it is of the form | y |. | z | | z |

  35. Another way to specify the range is to use the columns of the original matrix indicated by the stair step ones. | 4| | 7 | | 4s+7t | The range is the span of | 3| | 5 | or | 3s+5t | | 1| | 2 | | s+2t | But with this designation, it is harder to tell if something is in the range or not.

  36. Grading Scale • 93-100 A • 90-92 A- • 87-89 B+ • 83-86 B • 80-82 B- • 75-79 C+ • 70-74 C • 65-69 C- • 64 D+ • 61-63 D • 60 D-

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