1 / 20

Sec 4.2 + Sec 4.3 + Sec 4.4

CHAPTER 4 Vector Spaces. Sec 4.2 + Sec 4.3 + Sec 4.4. Vector Space. Set:. Let V be a set of elements with vector addition and multiplication by scalar is a vector space if these operations satisfy the following:. Vector Addition:. Scalar Multiplication:. Set:. Vector Addition:.

Download Presentation

Sec 4.2 + Sec 4.3 + Sec 4.4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Vector Space Set: Let V be a set of elements with vector addition and multiplication by scalar is a vector spaceif these operations satisfy the following: Vector Addition: Scalar Multiplication: Set: Vector Addition: Scalar Multiplication:

  2. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Vector Space Set: Let V be a set of elements with vector addition and multiplication by scalar is a vector spaceif these operations satisfy the following: Vector Addition: Scalar Multiplication: Set: Vector Addition: Scalar Multiplication:

  3. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Linear combination v is a linear combination of u1,u2

  4. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Linearly dependent vectors are said to be linearly dependent provided that one of them is a linear combination of the remaining vectors otherwise, they are linearly independent v is a linear combination of u1,u2 { u1, u2, v} are linearly dependent { f, g, h } are linearly dependent { f, g, h } are linearly dependent

  5. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Linearly dependent vectors

  6. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Wronskian Find the wroskian Find the wroskian Find the wroskian

  7. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Wronskian THM:

  8. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Definition: Subspace V W Let V be a set of elements with vector addition and multiplication by scalar is a vector spaceif these operations satisfy the following: W is a subspace of V provided that W itself is a vector space with addition operation and scalar multiplication as defined in V

  9. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 V THM: W Two conditions are satisfied W subspace of V

  10. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Spanning set span the vector space V if every vector in V is a a linear combination of these k-vectors Linearly Independent Linearly independent if the only solution for is Definition: is a basis for the vector space V if

  11. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Definition: is a basis for the vector space V if

  12. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 The dimension of a vector space V is the number of vectors in any basis of V Definition: Find dim(W)

  13. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 The dimension of a vector space V is the number of vectors in any basis of V Definition: Find dim(W)

  14. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 Homogeneous Linear System FACT: the solution set of Ax=0 is a subspace Consider the homogeneous linear system Consider the homogeneous linear system

  15. How to find a basis for the solution space W of the Homogeneous Linear System Ax=0 1 2 Consider the homogeneous linear system 3 4 5 6 7

  16. How to find a basis for the solution space W of the Homogeneous Linear System Ax=0 1 Consider the homogeneous linear system 2 3 4 5 6 7 dim( W ) = # of free variables = # columns A - # of leading variables

  17. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 n

  18. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 n

  19. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 n

  20. CHAPTER 4 Vector Spaces Sec 4.2 + Sec 4.3 + Sec 4.4 2 conditions out of 3

More Related