 Linear relation

A linear relation among vectors $v_1,\dots ,v_n$ is a coefficient tuple $(a_1,\dots ,a_n)$ satisfying ${\sum }_i{a}_i{v}_i=0$. It is nontrivial if at least one coefficient is nonzero. For the matrix with these vectors as columns, the space of relations is its kernel. The vectors are linearly independent exactly when this kernel is zero. An invertible change of coordinates preserves every relation, since $S{\sum }_i{a}_i{v}_i=0$ if and only if ${\sum }_i{a}_i{v}_i=0$.

A linear relation refers to a relationship between two variables where the change in one variable is proportional to the change in the other variable.