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Linear combination

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.

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Linear combination

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  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??

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