# 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.

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.