Linear combination

1. CHAPTER 4 Vector Spaces Sec 4.3 Linear combination 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

2. CHAPTER 4 Vector Spaces Sec 4.3 Linear combination 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

3. 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

4. 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:

5. 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

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

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

8. CHAPTER 4 Vector Spaces Sec 4.3 Linearly dependent vectors

9. 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

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

11. 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

12. 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

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

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

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

16. 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

17. 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??