Algebra by Michael Artin

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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Prove that, a matrix which has a row of zeros is not invertible. 16. A square matrix A is called nilpotent if A k = 0 for some k > O. Prove that if A is nilpotent, then / + A is invertible. 17. (a) Find infinitely many matrices B such that BA = /2 when (b) Prove that there is no matrix C such that AC = /3. Chapter 1 Exercises 33 18. 18) carefully, using the associative law to expand the product (AB)(B-1A- 1). 19. The trace of a square matrix is the sum of its diagonal entries: tr A = all + a22 + ...

6] is an element of order 6 in GL, (I\l), so the cyclic subgroup it generates has order 6. On the other hand, the matrix infinite order. because [~ [6 :] has :r [~ a = We may also speak of the subgroup of a group G generated by a subset U. This is the smallest subgroup of G containing U, and it consists of all elements of G which can be expressed as a product of a string of elements of U and of their inverses. In particular, a subset U of G is said to generate G if every element of G is such a product.

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'. We could also write this condition as (ab)' = a'b ' . Of course, the choice of G as domain for this isomorphism is arbitrary.

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