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Transcript
1.7: Linear Independence We will study homogeneous equations from a different perspective by writing them as vector equations. In section 1.5 we focused on the unknown solutions to Ax 0 In section 1.7 will focus on the vectors that appear in the vector equation corresponding to Ax 0 . Consider the matrix equation, Ax 0 , where 3 5 4 A 3 2 4 6 1 8 3 5 4 x1 0 So 3 2 4 x2 0 has the equivalent vector 6 1 8 x3 0 equation form: 3 5 4 0 x1 3 x2 2 x3 4 0 6 1 8 0 Note: When x1 x2 x3 0 we have the trivial solution. Are there other solutions? Definitions v , v ,, v in 1 2 p n is said to be linearly independent if x1 v1 x2 v2 x p v p 0 has only the trivial solution. v , v ,, v 1 2 p in is said to be linearly dependent if there exist weights n c1 , c2 , , c p , not all zero, such that c1 v1 c2 v2 c p v p 0 Linear Independence v , v ,, v 1 2 p n in are linearly independent x1 v1 x2 v2 x p v p 0 has only the trivial solution. x1 v1 x2 v2 x p v p 0 x1 x2 x p 0 Examples: Determine each of the following sets of vectors are linearly independent. 1 v2 0 1. 1 v1 2 2. 1 v1 0 2 v2 0 3. 1 v1 2 1 v2 0 0 v3 1 Find a linear dependence relationship among the vectors: 1 1 v1 v2 2 0 0 v3 1 The columns of a matrix A are linearly independent Ax 0 has only the trivial solution. Examples: Determine if the columns of the following matrices are linearly independent. 1 2 1. 3 4 1 2 2. 2 4 Tips to determine the linear dependence A set of vectors are linearly dependent if any of the following are true: 1. The set has two vectors and one is a multiple of the other. 2. The set has two or more vectors and one of the vectors is a linear combination of the others. 3. The set contains more vectors than the number of entries in each vector. 4. The set contains the zero vector.