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.