linearly dependent

Linear independence and determinants. We can use (10) to see that
(11) the rows of a 3 × 3 matrix A are dependent � det A = 0.
Proof. If we denote the rows by a1,a2, and a3, then from 18.02, volume of the parallelepiped = a1 · (a2 × a3) = detA,
spanned by a1, a2, a3 so that a1, a2, a3 lie in a plane � detA =0.
The above statement (11) generalizes to an n × n matrix A ; we rephrase it in the statement below by changing both sides to their negatives. (We will not prove it, however.)