  We provide a doubly exponential upper bound in $p$ on the size of forbidden
pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field
$\mathbb{F}$ of linear rank-width at most $p$. As a corollary, we obtain a
doubly exponential upper bound in $p$ on the size of forbidden vertex-minors
for graphs of linear rank-width at most $p$. This solves an open question
raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear
rank-width at most $k$. European J. Combin., 41:242--257, 2014]. We also give a
doubly exponential upper bound in $p$ on the size of forbidden minors for
matroids representable over a fixed finite field of path-width at most $p$.
  Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on
the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series
B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded
path-width. To adapt this notion into linear rank-width, it is necessary to
well define partial pieces of graphs and merging operations that fit to
pivot-minors. Using the algebraic operations introduced by Courcelle and
Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we
define boundaried $s$-labelled graphs and prove similar structure theorems for
pivot-minor and linear rank-width.
