Linear Independence

Given a set if vectors $\{{\text{v}}_1,{\text{v}}_2,\dots ,{\text{v}}_k\}$

They are linearly independent if: $c_1{\text{v}}_1,c_2{\text{v}}_2,\dots ,c_k{\text{v}}_k=\text{0}$ Only has trivial solution. Else they are linearly dependent On the flip side, if: $c_1{\text{v}}_1,c_2{\text{v}}_2,\dots ,c_k{\text{v}}_k=\text{0}$ And a nontrivial solution ($$\begin{gathered} c_1,c_2,\dots ,c_k\ne \left[\text{matrix}0 \\ ⋮ \\ 0\right] \end{gathered}$$) solution exists, they are linearly dependent.

What is the smallest number of vectors needed in a parametric solution to a linear system?

Vectors are dependent if:

• One of the vectors is a 0 vector.
• They are in each others Span ($c{v}_1=v_2\text{When }c\ne 0$)

Theorems

• More Vectors Than Elements: The vectors must be linearly dependent as column cannot be pivotal so there are free variables.
• Set Contains Zero Vector: If one of the vectors is a 0 vectors the system is linearly dependent