By Marvin Marcus

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**Extra resources for Introduction to Linear Algebra**

**Sample text**

The integration theory in which the group is constructed from the algebra is due to Lie himself and was a real tour de force in the development of the subject. Ado's theorem on the other hand, states that every Lie algebra is isomorphic to a matrix Lie algebra; and this fact obviates Lie's original method. Given any Lie algebra we may construct a representation ofit by a linear Lie algebra and so every Lie group is locally isomorphic to a matrix group. This situation, however, does not in general hold in the large (see G.

The latter is uniquely given by an orthogonal transformation O(t) which relates a fixed inertial frame e1, e2 , e3 to a moving frame E1(t), E 2 (t), E3(t) attached to the body. The configuration space of the body may therefore be identified with the group of rigid motions in [R3. More generally, problems of describing the motion of a dynamical system in a moving coordinate system arise, for example, in meteorology or oceanography, where the earth's rotation introduces centrifugal and Coriolis forces into the equations of motion.

We shall proceed heuristically and construct a Hilbert space under the (plausible) assumption that H has an eigenvector t/I).. (The plausibility of this assumption depends on whether you are a mathema- 15. ) We will see later that the Hilbert space we construct is 12 , the space of square summable sequences. If H 1/1;. = 2hwl/l;. then the commutation relation [H, a] = - hwa shows that al/l;. ) = (aH hwa) 1/1 ;. = a(H - hw)I/I;. ). Similarly, [H, a*] = hwa* and so a*I/I;. is an eigenvector of H with eigenvalue (2 + l)hw.