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.
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.
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.
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.