Sec 4.2 + Sec 4.3 + Sec 4.4

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