By Michael Artin

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

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**Example text**

AJ and [ai+l .. , an] are formed first and the results are then multiplied using the given law of composition. Proof. We use induction on n. The product is defined by (i) and (ii) for n <: 2, and it does satisfy (iii) when n = 2. Suppose that we know how to define the product of r elements when r <: n - 1, and that this product is the unique product satisfying (iii). We then define the product of n elements by the rule Cal ... a,,] Cal ... an-l][an], = where the terms on the right side are those already defined.

If A is invertible we can multiply both sides of the equation AX = 0 by A-I to derive X = O. So the equation AX = 0 has only the trivial solution. This shows that (c) implies (d). To prove the last implication, that (d) implies (a), we take a look at square row echelon matrices M. 19) Let M be a square row echelon matrix. Either M is the identity matrix, or its bottom row is zero. 15). Suppose that (a) does not hold for a given matrix A. Then A can be reduced by row operations to a matrix A' whose bottom row is zero.

It is a subgroup of G-the smallest subgroup which contains x. 7) correctly, we must remember that x n is a notation for a certain element of G. It may happen that there are repetitions in the list. For example, if x = 1, then all elements in the list are equal to 1. We may distinguish two possibilities: Either the powers of Section 2 47 Subgroups x are all distinct elements, or they are not. In the first case, the group H is called infinite cyclic. Suppose we have the second case, so that two powers are equal, say x n = x m , where n > m.