"Hermann Graßmann submitted [Die lineale Ausdehnungslehre] as a Ph. D. thesis, but Möbius said he was unable to evaluate it and forwarded it to Ernst Kummer, who rejected it without giving it a careful reading."
From a webpage : "His 1844 work Die lineale Ausdehnungslehre: Ein neuer Zweig der Mathematik [The Theory of Linear Extension, a New Branch of Mathematics], effectively single-handedly founded Linear Algebra. This work was submitted as a Ph.D. thesis in Mathematics, however its formulation of linear vector space in opposition to the canonical Euclidean geometry of the time was too radical for his contemporary mathematics establishment and was rejected. Consequently, the significance of contribution to the mathematical sciences were largely unrecognized in his lifetime, but they were eventually re-discovered towards the end of the nineteenth century and early twentieth century. It was because of these rejections and early lack of recognition of his work in the mathematics that he suffered he turned himself to Vedic studies, and made those discoveries in there which we know him best for [!]" -- Hermann Graßmann: Philologist and Mathematician
The claim about "single-handedly founding linear algebra" seems exaggerated. For a closer investigation, one might e.g. look at the articles by D. Fearnley-Sander quoted and referenced in the wikipedia article.
Added: After reading (small) parts of Engel's biography (vol. III.2 in Graßmann's Gesammelte Werke), I feel that the quotations above are kind of unfair towards Moebius and Kummer. Moebius actually put quite an effort into supporting Graßmann, for several years. Regarding Kummer, it should first of all be noted that Graßmann did not submit his work as a Ph.D. thesis in the modern sense, but sent it, along with another work, to the ministry, to apply for a professorship at some university. Kummer's report (reprinted there, pp. 126--129) is ambiguous, in that he harshly criticises the form, but admits that "diese Schrift wirklich neue und interessante Gesichtspunkte gewährt, so daß ich über den wissenschaftlichen Wert des Inhalts mich wirklich lobend und anerkennend äußern kann". Kummer indicates that more profound results may be found in Graßmann's work with more effort. But for a teaching position, he suggests, there are younger excellent mathematicians with much better style of exposition. He also says that he has no reservations against awarding Graßmann the title "Professor", but given Graßmann's deficits in exposition, he has doubts about him as lecturer; however, he says, it could still be enquired whether his oral teaching abilities are better (given that Graßmann was a school teacher). Apparently, the ministry did not take the last suggestion serious, but wrote to some lower school office whether it would be OK to award Graßmann solely the title; the office advised against this, saying it would cause trouble with superior teachers who did not have this title (...). So the ministry wrote back to Graßmann, dismissing his wishes. -- Frustrating as this must have been for Graßmann, but given Peter Michor's comment, I agree that many statements in Joël 's answer do apply very much to Graßmann's case.