linear dependence

thus, a set of vectors is linearly dependent if and only if one of them is zero or a linear combination of the others.

a set of vectors $\text{[math]}$ is linearly independent if the only solution to
$\text{[math]}$ an equivalent definition is that vectors $\text{[math]}$ are linearly independent if and only if a vector $\text{[math]}$ can be written as a linear combination of those vectors in at most one way:
$\text{[math]}$ [cite:;taken from @calc_hubbard_2015 definition 2.4.2]

let $\text{[math]}$ be an $\text{[math]}$ matrix. then the number of linearly independent columns of $\text{[math]}$ equals the number of linearly independent rows.
[cite:;taken from @calc_hubbard_2015 proposition 2.5.11]

a set of vectors is just a subset of a vector space $\text{[math]}$ while a sequence of vectors is a map $\text{[math]}$ (can also be written as a infinite tuple). a set does not care about ordering or enumerating elements multiple times in contrast to a sequence.

$\text{[math]}$ is a linearly dependant set if and only if $\text{[math]}$

first we prove
$\text{[math]}$ $\text{[math]}$ being linearly dependant means there exists $\text{[math]}$ such that:
$\text{[math]}$ we multiply both sides of this equation by $\text{[math]}$
$\text{[math]}$ second we prove
$\text{[math]}$ we need to find $\text{[math]}$ such that
$\text{[math]}$ and that $\text{[math]}$ can just be $\text{[math]}$ which would give us $\text{[math]}$ , therefore $\text{[math]}$ is linearly dependant