Expalining Linear Dependence of Vectors

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.
  
  Courtesy: Stat Trek