# Definition: Linearly Dependent and Linearly Independent Vectors, Zero Vector

Let $$V$$ be a vector space over a field $$F$$. Finitely many vectors $$x_1,\ldots,x_n\in V$$ are called linearly dependent, if there exist field elements $$\alpha_1,\ldots,\alpha_n\in F$$, which are not all equal zero, for which $\alpha_1 x_1 + \ldots \alpha_n x_n=0,$ i.e. for which the corresponding linear combination of the vectors $$x_1,\ldots,x_n\in V$$ is the zero vector, although $$\alpha_i\neq 0$$ for at least one $$i=1,\ldots, n$$.

The finitely many vectors $$x_1,\ldots,x_n\in V$$ are called linearly independent, if they are not linearly dependent, i.e. if from $\alpha_1 x_1 + \ldots \alpha_n x_n=0$ it always follows that $$\alpha_1=\ldots=\alpha_n=0$$, which means that the zero vector has only a trivial linear combination of the $$x_1,\ldots,x_n\in V$$ with all scalars $$\alpha_i\in F$$ being zero.

Definitions: 1 2 3
Lemmas: 4
Parts: 5
Proofs: 6 7 8

Github: ### References

#### Bibliography

1. Koecher Max: "Lineare Algebra und analytische Geometrie", Springer-Verlag Berlin Heidelberg New York, 1992, 3rd Volume