Linearly Dependent Sequence of Vector Space

Theorem
Let $\struct {G, +}$ be a group whose identity is $\mathbf 0$.

Let $\struct {G, +, \circ}_K$ be a $K$-vector space.

Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of distinct non-zero vectors of $G$.

Then $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ is linearly dependent :
 * $\exists p \in \closedint 2 n: a_p$ is a linear combination of $\sequence {a_k}_{1 \mathop \le k \mathop \le p - 1}$

Necessary Condition
Suppose $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ is linearly dependent.

By hypothesis, the set of all integers $r \in \closedint 1 n$ such that $\sequence {a_k}_{1 \mathop \le k \mathop \le r}$ is linearly dependent is not empty.

Let $p$ be its smallest element.

Then from Singleton is Linearly Independent‎, $p \ge 2$, as $a_1 \ne \mathbf 0$ and hence $\set {a_1}$ is linearly independent.

Also, there exist scalars $\lambda_1, \ldots, \lambda_p$, not all of which are zero, such that $\displaystyle \sum_{k \mathop = 1}^p \lambda_k \circ a_k = \mathbf 0$.

Suppose $\lambda_p = 0$.

Then not all of $\lambda_1, \ldots, \lambda_{p-1}$ can be zero.

Then $\sequence {a_k}_{1 \mathop \le k \mathop \le p-1}$ is linearly dependent.

That contradicts the definition of $p$, so $\lambda_p \ne 0$.

So, because:
 * $\displaystyle \lambda_p \circ a_p = - \sum_{k \mathop = 1}^{p - 1} \lambda_k \circ a_k$

we must have:
 * $\displaystyle a_p = \sum_{k \mathop = 1}^{p - 1} \paren {-{\lambda_p}^{-1} \lambda_k} \circ a_k$

and thus $a_p$ is a linear combination of $\sequence {a_k}_{1 \mathop \le k \mathop \le p - 1}$.

Sufficient Condition
Now suppose that $a_p$ is a linear combination of $\sequence {a_k}_{1 \mathop \le k \mathop \le p - 1}$.

Then:
 * $\displaystyle a_p = \sum_{k \mathop = 1}^{p - 1} \mu_k \circ a_k$

So we can assign values to $\lambda_k$ as follows:


 * $\forall k \in \closedint 1 n: \lambda_k = \begin{cases}

\mu_k & : k < p \\ -1 & : k = p \\ 0 & : k > p \\ \end{cases}$

Then:
 * $\displaystyle \sum_{k \mathop = 1}^n \lambda_k \circ a_k = \mathbf 0$

Hence the result.