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
   =    2 
1
2
   +    3 
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.