**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** = 2**a**.
- 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

## No comments:

Write comments