By Michael Artin

This booklet is for the honors undergraduate or introductory graduate path. Linear algebra is tightly built-in into the textual content.

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Suppose, for example, that each matrix entry au is a differentiable function of a single variable: ai) = aij(t). Then det A is also a differentiable function of t, because sums and products of differentiable functions are differentiable. 5. CRAMER'S RVLE The name Cramer's Rule is applied to a group of formulas giving solutions of systems of linear equations in terms of determinants. To derive these formulas we need to use expansion by minors on columns other than the first one, as well as on rows.

For example, the symmetric group Sn of permutations of {I, ... ,D} is isomorphic to the group of permutation matrices, a subgroup of GL n (~), and we often blur the distinction between these two groups. 1) asymmetrically as a function, or map cp: G~ G'. Thus an isomorphism cp from G to G' is a bijective map which is compatible with the laws of composition. 2) cp (ab) = cp(a)cp (b), for all a, bEG. The left side of this equality means to multiply a and b in G and then apply cp, while on the right the elements cp(a) and cp(b), which we denoted by a', b' before, are multiplied in G'.

The set S of integers n such that x n = 1 is a subgroup of 7L+. Proof. If x m = 1 and x n = 1, then x m + n = x m x n = 1 too. This shows that m + n E S if m, n E S. So axiom (i) for a subgroup is verified. Also, axiom (ii) holds because XO = 1. Finally, if x n = 1, then x- n = xnx- n = Xo = 1. Thus - n E S if n E S. 8) Lemma. 4) that S = m7L, where m is the smallest positive integer such thatx m = 1. The m elements 1, x, ... ,x m - l are all different. (If Xi = xi with 0 <: i < j < m, then x i - i = 1.