Linear Dependence

Definition: Linear Dependence

Let ${\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n$ be vectors in some vector space $(V,F,+,\cdot$).

We say that ${\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n$ are linearly dependent iff they are not linearly independent, i.e. there exist $c_1,\cdots ,c_n$ with at least one $c_i\ne 0$ such that

$$c_1{\mathbf{v}}_1+\cdots +c_n{\mathbf{v}}_n=0$$

Theorem: Linear Dependence $⟹$ Linear Combination

If $\{{\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n\}(n\ge 2)$ are linearly dependent vectors, then there is at least one ${\mathbf{v}}_i\in \{{\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n\}$ which can be expressed as a linear combination of the rest of the vectors:

$${\mathbf{v}}_i=\sum _{\begin{array}{rl}j& =1\\ j& \ne i\end{array}}^n{c}_j{\mathbf{v}}_j$$

PROOF

According to the definition of linear dependence, there are coefficients $h_1,\cdots ,h_n\in F$ with at least one $h_i\ne 0$ such that

$$h_1{\mathbf{v}}_1+\cdots +h_i{\mathbf{v}}_i+\cdots +h_n{\mathbf{v}}_n=0$$

Let’s move everything except $h_i{\mathbf{v}}_i$ to the other side of the equation:

$$h_i{\mathbf{v}}_i=-h_1{\mathbf{v}}_1+\cdots -h_{i-1}{\mathbf{v}}_{i-1}-h_{i+1}{\mathbf{v}}_{i+1}+\cdots +-h_n{\mathbf{v}}_n$$

Since $h_i\ne 0$, we can divide both sides by it:

$${\mathbf{v}}_i=c_1{\mathbf{v}}_1+\cdots +c_{i-1}{\mathbf{v}}_{i-1}+c_{i+1}{\mathbf{v}}_{i+1}+\cdots +c_n{\mathbf{v}}_n$$

We have thus obtained ${\mathbf{v}}_i$ as a linear combination of the other vectors, where $c_j=-\frac{h_j}{h_i}$