Stat Trek

Teach yourself statistics

Stat Trek

Teach yourself statistics

Show navigation menu Search site

Stat Trek

Teach yourself statistics

Show navigation menu Search site

Independent vs. Dependent Vectors

One vector is dependent on other vectors, if it is a linear combination of the other vectors.

Linear Combination of Vectors

If one vector is equal to the sum of scalar multiples of other vectors, it is said to be a linear combination of the other vectors.

For example, suppose a = 2b + 3c, as shown below.

11
16
   =   
1
2
   +   
3
4
   =   
2*1 + 3*3
2*2 + 3*4
a b c 2b + 3c

Note that 2b is a scalar multiple and 3c is a scalar multiple. Thus, a is a linear combination of b and c.

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.

Test Your Understanding

Problem 1

Consider the row vectors shown below.

0 1 2
3 2 1
a b

3 3 3
3 4 5
c d

Which of the following statements are true?

(A) Vectors a, b, and c are linearly dependent.
(B) Vectors a, b, and d are linearly dependent.
(C) Vectors b, c, and d are linearly dependent.
(D) All of the above.
(E) None of the above.

Solution

The correct answer is (D), as shown below.

  • Vectors a, b, and c are linearly dependent, since a + b = c.
  • Vectors a, b, and d are linearly dependent, since 2a + b = d.
  • Vectors b, c, and d are linearly dependent, since 2c - b = d.