By George Boole

This Elibron Classics e-book is a facsimile reprint of a 1877 version through Macmillan and Co., London.

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Suppose that L ∈ GL(n, R). Then L preserves the dot product on Rn if and only if L−1 = LT , the transpose of the matrix L. ) Proof. Recall that the transpose of a matrix is obtained by reflecting across the main diagonal: (LT )ij := Lji . We first show that for all v, w ∈ Rn , we have Lv · w = v · LT w. Indeed, note that if we think of v and w as column vectors, then wT is a row vector, and we may express the dot product as matrix multiplication: v · w = wT v. Replacing v by Lv yields Lv · w = wT (Lv) = (wT L)v = (LT w)T v = v · LT w.

Then the velocity of the particle is given by the timederivative r˙ , and its linear momentum is defined to be p := m˙r. Thus, the linear momentum is a measure of the particle’s linear motion, taking into account both its velocity and mass. 1). Now let’s revisit the top, T , that observer P started spinning at the end of the previous chapter. 1: The position, linear momentum, and angular momentum of a particle moving in three-dimensional space. 17 18 Symmetry and Quantum Mechanics makes a full revolution every 2π/ω seconds.

Show that the n function ϕ : X → R defined by ϕ(x) := (a1 , a2 , . . , an ) is an isomorphism of vector spaces. Further, show that ϕ preserves the inner products, hence is an isomorphism of inner product spaces: for all x1 , x2 ∈ X, we have x1 , x2 = φ(x1 ) · φ(x2 ). Thus, once P chooses a positively oriented orthonormal basis, he has identified space with (R3 , ·) endowed with the familiar right-handed orientation in which the standard basis is positive. With this, P feels like he is back on firm ground and starts thinking about some experiments he wants to perform.