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
