CHAPTER 4 Vector Spaces Linear combination Sec 4.3 IF The vector w is called a linear combination of the vectors IF there exists scalars such that Find echelon of: Determine whether the vector w is a linear combination of the vectors Determine whether the vector w is a linear combination of the vectors Determine whether the vector w is a linear combination of the vectors No sol consistent Not linear combination linear combination

CHAPTER 4 Vector Spaces Linear combination Sec 4.3 IF The vector w is called a linear combination of the vectors IF there exists scalars such that v is a linear combination of u1,u2

Linearly dependent vectors CHAPTER 4 Vector Spaces Sec 4.3 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 { f, g, h } are linearly dependent { f, g, h } are linearly dependent Determine whether the vector w is a linear combination of the vectors v is a linear combination of u1,u2 { u1, u2, v} are linearly dependent

Space spanned by vectors CHAPTER 4 Vector Spaces Sec 4.3 Space spanned by vectors W = set of all possible linear combination of the vectors We say W is spanned by the vectors v1 and v2 {v1, v2} is called the spanning set for W. THM1:

Linearly dependent vectors CHAPTER 4 Vector Spaces Sec 4.3 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 Linearly independent vectors are said to be linearly independent provided that the equation: has only the trivial solution

Linearly dependent vectors CHAPTER 4 Vector Spaces Sec 4.3 Linearly dependent vectors We solve: homog echelon Only the trivial solution

Linearly dependent vectors CHAPTER 4 Vector Spaces Sec 4.3 Linearly dependent vectors We solve: homog WHY? Inf. sol Remark

Linearly dependent vectors CHAPTER 4 Vector Spaces Sec 4.3 Linearly dependent vectors

Linearly independent vectors CHAPTER 4 Vector Spaces Sec 4.3 Linearly independent vectors are said to be linearly independent provided that the equation: has only the trivial solution Linearly dependent vectors There exists scalars not all zeros such that linearly dependent

Linearly dependent vectors CHAPTER 4 Vector Spaces Sec 4.3 Linearly dependent vectors Only the trivial solution Remark

Linearly dependent vectors CHAPTER 4 Vector Spaces Sec 4.3 Linearly dependent vectors There exists scalars not all zeros such that linearly dependent At least one of them is a linear combination of the others linearly dependent

TH2: independence of n vectors in R^n CHAPTER 4 Vector Spaces Sec 4.3 TH2: independence of n vectors in R^n The matrix linearly independent has nonzero determinant Columns of the matrix A linearly independent linearly dependent

TH2 This is true only for n vectors in R^n

NOTE: Def or th3 ?????? Use determinant

Theorem7:(p193) A All statements are equivalent row equivalent Ax = b Columns of are linearly independent row equivalent Ax = b Every n-vector b has unique sol is a product of elementary matrices Ax = b Every n-vector b is consistent Ax = 0 The system has only the trivial sol nonsingular All statements are equivalent

TH3: Fewer than n vectors in R^n CHAPTER 4 Vector Spaces Sec 4.3 TH3: Fewer than n vectors in R^n The matrix nxk Some kxk submatrix of A has nonzero determinant linearly independent

TH3: Fewer than n vectors in R^n CHAPTER 4 Vector Spaces Sec 4.3 TH3: Fewer than n vectors in R^n The matrix nxk Some kxk submatrix of A has nonzero determinant linearly independent Size??? How many submatrix??