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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24c and d), we can deduce facts about right multiplication, XP, from the corresponding facts about left multiplication. 25) (a) Add a . (column i) to (column j). (b) Interchange (column i) and (column j). (c) Multiply (column i) by c O. "* 3. DETERMINANTS Every square matrix A has a number associated to it called its determinant. In this section we will define the determinant and derive some of its properties. The determinant of a matrix A will be denoted by det A. 2) ~] det [: = ad - be. If we think of a 2 x 2 matrix A as an operator on the space (R2 of real twodimensional vectors, as in Section 2, then det A can be interpreted geometrically.

Is of the type H = bll.. for some integer b. 4) Proposition. Proof. We leave the verification that bll.. is a subgroup as an exercise and proceed to show that every subgroup has this form. +. + is addition, the identity element is 0, and the inverse of a is -a, So the axioms for a subgroup read (i) if a E Hand b E H, then a (ii) 0 E H; (iii) if a E H, then -a E H. + b E H; By axiom (ii), 0 E H. , so that case is settled. If not, there is a positive integer in H. For let a E H be any nonzero element.

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. The inverse function cp-]: G' ~ G would serve just as well. Two groups G and G' are called isomorphic if there exists an isomorphism cp: G ~ G '.