For the purposes of the theory the complex vector space VC, obtained by extending the scalars of V from Q to C, is used.
Once formulated in this fashion, the rational representation ρ of T on V setting up the Hodge structure F determines the image ρ(U(1)) in GL(VC); and MT(F) is by definition the smallest algebraic group defined over Q containing this image.
[1] The original context for the formulation of the group in question was the question of the Galois representation on the Tate module of an abelian variety A. Conjecturally, the image of such a Galois representation, which is an l-adic Lie group for a given prime number l, is determined by the corresponding Mumford–Tate group G (coming from the Hodge structure on H1(A)), to the extent that knowledge of G determines the Lie algebra of the Galois image.
A related conjecture on abelian varieties states that the period matrix of A over number field has transcendence degree, in the sense of the field generated by its entries, predicted by the dimension of its Mumford–Tate group, as in the previous section.
Work of Pierre Deligne has shown that the dimension bounds the transcendence degree; so that the Mumford–Tate group catches sufficiently many algebraic relations between the periods.