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.

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Sec Sec 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:

Sec Sec 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:

Linear combination v is a linear combination of u1,u2 Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces

Linearly dependent vectors are said to be linearly dependent provided that one of them is a linear combination of the remaining vectors Sec Sec 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

Linearly dependent vectors Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces

Wronskian Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces Find the wroskian

Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces THM: Wronskian

Sec Sec 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

Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces THM: V W W subspace of V Two conditions are satisfied

Spanning set Sec Sec 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

Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces Definition: is a basis for the vector space V if

Sec Sec 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)

Sec Sec 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)

Sec Sec 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

How to find a basis for the solution space W of the Homogeneous Linear System Ax=0 Consider the homogeneous linear system

How to find a basis for the solution space W of the Homogeneous Linear System Ax=0 Consider the homogeneous linear system dim( W ) = # of free variables = # columns A - # of leading variables

Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces n

Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces n

Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces n

Sec Sec Sec 4.4 CHAPTER 4 Vector Spaces 2 conditions out of 3