By Michael Artin

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

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**Sample text**

When one speaks of classifying groups, what is meant is to describe the isomorphism classes. 1)]. A confusing point about isomorphisms is that there exist isomorphisms from a group G to itself: cp: G~G. Such an isomorphism is called an automorphism of G. The identity map is an automorphism, of course, but there are nearly always other automorphisms as well. For example, let G = {1, x, x 2 } be a cyclic group of order 3, so that x 3 = 1.

Begin by finding a nice solution to the matrix equation A 2 = -I. 3. (Vandermonde determinant) (a) Prove that det[~ ib ~ J a 2 2 c = (b - a)(c - a)(c - b). 2 *(b) Prove an analogous formula for n x n matrices by using row operations to clear out the first column cleverly. *4. Consider a general system AX = B of m linear equations in n unknowns. If the coefficient matrix A has a left inverse A I , a matrix such that AIA = In, then we may try to solve the system as follows: AX = B A'AX x = A'B = A'R.

3) ~ i / / / / / ~I / 1/ ~ ~ '" _ Matrix Operation 20 Chapter 1 For example, if A = 103] [o 2 1 5 2 , then 1 A2I [0 3] = 5 . 4) det A = all det All - a21 det A 21 +, - ... + ani det AnI. The signs alternate. 1), as a recursive definition of the determinant. 2) for 2 X 2 matrices. de{ ~ ~J. 2), to get detA = 1'(-9) - 2ยท(-15) + 0'(-3) = 21. 2)]' It is important, both for computation of determinants and for theoretical considerations, to know some of the many special properties satisfied by determinants.