Linear Dependency

Lets look again at two of the vectors we saw in the previous quiz:

• \vec{v_2}=\begin{bmatrix} 2\\ 2\\ 2\end{bmatrix}

  • \vec{v_3}=\begin{bmatrix} 8\\ 8\\ 8\end{bmatrix}

We stated that one vector can be derived from the other by a simple mathematical linear combination.

For example:

4\vec{v_2} =\vec{v_3}

When one vector can be defined as a linear combination of the other vectors, they are a set of linearly dependent vectors.

When each vector in a set of vectors vector can not be defined as a linear combination of the other vectors, they are a set of linearly independent vectors.

In our example,

• { \vec{v_2} , \vec{v_3} } is a linearly dependent set

• { \vec{v_1} , \vec{v_2} } is a linearly independent set

and

• { \vec{v_1} , \vec{v_3} } is a linearly independent set

( \vec{v_1} is defined in the previous quiz as: \begin{bmatrix} 1\\ 2\\ 3\end{bmatrix} )

The easiest way to know if a set of vectors is linear dependent or not, is with the use of determinants .
Determinants are beyond the scope of our Linear Algebra Essentials and we will not focus on that.