MATH1231 - Linear Independence

The goal of independence is determining whether we can find a spanning set for a given subspace that is 'minimal' (i.e. each 'building block' is non-redundant, or independent)

Linearly Dependent Example
(1 1) (2 2) can be written as (1 1) 2(1 1), and the two are hence not linearly dependent.

Linearly Independent Example
(1 1) (2 3) are not dependent as one cannot be written as a scalar multiple of the other.

For S = {v1, v2, … , vn}

If we can find scalars (that aren't all zero) so that

α1v1 + α2v2 + … + αnvn = 0

then S is linearly dependent

Else if the only solution is all α being 0

then S is linearly independent

If all the vectors are linearly independent then the dim(span{v1, v2, …, vn}) = the number of vectors.