1. Sec 4.2 + Sec 4.3 + Sec 4.4 CHAPTER 4 Vector Spaces 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 Space Set: Vector Addition: Scalar Multiplication: Set: Vector Addition: Scalar Multiplication:

2. Sec 4.2 + Sec 4.3 + Sec 4.4 CHAPTER 4 Vector Spaces 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 Space Set: Vector Addition: Scalar Multiplication: Set: Vector Addition: Scalar Multiplication:

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

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

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

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

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

8. Sec 4.2 + Sec 4.3 + Sec 4.4 CHAPTER 4 Vector Spaces Let V be a set of elements with vector addition and multiplication by scalar is a vector space if these operations satisfy the following: Subspace Definition: V W 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. Sec 4.2 + Sec 4.3 + Sec 4.4 CHAPTER 4 Vector Spaces THM: V W W subspace of V Two conditions are satisfied

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

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

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

14. Sec 4.2 + Sec 4.3 + Sec 4.4 CHAPTER 4 Vector Spaces FACT: the solution set of Ax=0 is a subspace 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 Consider the homogeneous linear system 1 2 3 5 4 6 7

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

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

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

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

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