Linear & multilinear algebra; matrix theory
Matrices, vector spaces, linear transformations, tensors.
Founded: 1858 — Cayley's A Memoir on the Theory of Matrices
The claim
No credible dismissal found. The famous line calling matrix algebra “an algebra of no conceivable use” could not be verified after an extensive search (including the standard scholarly history, Higham 2008) and looks apocryphal. A genuine, different Hardy quote exists: A Mathematician's Apology (1940) notes “no one foresaw the applications of matrices and groups… to modern physics” — a retrospective note that applications were unforeseen, not a claim they'd never come.
What happened
Matrix mechanics (1925) — Heisenberg, formalized by Born & Jordan — one formulation of quantum mechanics.
Google PageRank (1998) — The principal eigenvector of the web link matrix.
67 years — founding (1858) to matrix mechanics (1925)
Verdict: Never dismissed, applied anyway.
Confidence: high.