Section 7 Definitions

Linear Independence

The vectors $\vec{v_1}, \vec{v_2}, ... ,\vec{v_n}$ in $\Re^n$ are linearly independent if the vector equation

(1)
\begin{align} x_1\vec{v_1} + x_2\vec{v_2} + ... + x_n\vec{v_n} = \vec{0} \end{align}

has only a trivial ($x = 0$) solution.

In summary, a linearly independent solution has only one solution, where each vector supplies independent information about the solution.
Two vectors that are not scalar multiples of eachother must be linearly independent.

If the vectors are linearly dependent, there must exist a set $c_1, c_2, c_3, ... , c_p$ where at least one coefficient is not zero such that

(2)
\begin{align} c_1\vec{v_1} + c_2\vec{v_2} + ... + c_p\vec{v_p} = \vec{0} \end{align}

In other words, there is more than just the zero solution. Ergo, if there are infinite solutions, it's linearly dependent.
If, however, there are only two vectors, they are linearly dependent if one is a scalar multiple of another.

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