Statistics Dictionary

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Linear Dependence of Vectors

A set of vectors is linearly independent if no vector in the set is (a) a scalar multiple of another vector in the set or (b) a linear combination of other vectors in the set; conversely, a set of vectors is linearly dependent if any vector in the set is (a) a scalar multiple of another vector in the set or (b) a linear combination of other vectors in the set.

Consider the row vectors below.

a =  
1 2 3
  d =  
2 4 6
b =  
4 5 6
  e =  
0 1 0
c =  
5 7 9
  f =  
0 0 1

Note the following:

  • Vectors a and b are linearly independent, because neither vector is a scalar multiple of the other.

  • Vectors a and d are linearly dependent, because d is a scalar multiple of a; i.e., d = 2a.

  • Vector c is a linear combination of vectors a and b, because c = a + b. Therefore, the set of vectors a, b, and c is linearly dependent.

  • Vectors d, e, and f are linearly independent, since no vector in the set can be derived as a scalar multiple or a linear combination of any other vectors in the set.

See also:  Vector Dependence | Matrix Rank