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XIII

Matrices and Linear Maps

Presumably reader s of this chapter will have had some basic acquaintance with linear algebra in elementary courses . We go beyond such courses by pointing out that a lot of results hold for free module s over a commutative ring. This is useful when one wants to deal with familie s of linear maps , and reduction modulo an ideal. Note that §8 and §9 give examples of group theory in the context of linear groups. Throughout this chapter, we let R be a commutative ring, and we let E, F be R -modules. We suppress the prefix R in front of linear maps and modules.

§1.

MATRICES

By an m x n matrix in R one means a doubly indexed family of elements of R, (ai) , (i = 1, . . . , m and j = 1, . . . , n), usually written in the form

We call the elements a., the coefficients or components of the matrix . A 1 x n matrix is called a row vector (of dimension, or size, n) and a m x 1 matrix is called a column vector (of dimension, or size, m). In general, we say that (m, n) is the size of the matrix, or also m x n. We define addition for matrices of the same size by components. If A = (ai) and B = (b i) are matrices of the same size, we define A + B to be the matrix whose (i-component is a ij + bi j • Addition is obviously associative. We define the multiplication of a matrix A by an element C E R to be the matrix (cai),

503 S. Lang, Algebra © Springer Science+Business Media LLC 2002

504

XIII, §1

MATRICES AND LINEAR MAPS

whose ij-component is caij' Then the set of m x n matrices in R is a module (i.e. an R-module). We define the product AB of two matrices only under certain conditions. Namely, when A has size (m, n) and B has size (n, r) , i.e. only when the size of the rows of A is the same as the size of the columns of B. If that is the case, let A = (aj) and let B = (b jk). We define AB to be the m x r matrix whose ikcomponent is n

L aijbjk. j= 1

If A , B, C are matrices such that AB is defined and BC is defined, then so is (AB)C and A(BC) and we have (AB)C = A(BC).

This is trivial to prove. If C = (Ck'), then the reader will see at once that the ii-component of either of the above products is equal to

L L aijbjkck/' j

k

An m x n matrix is said to be a square matrix if m = n. For example, a 1 x 1 matrix is a square matrix, and will sometimes be identified with the element of R occurring as its single component.

For a given integer n

~

1 the set of square n x n matrices forms a ring.

This is again trivially verified and will be left to the reader. The unit element of the ring of n x n matrices is the matrix

0 ... 0 0 0 1 In = 0

0

whose components are equal to 0 except on the diagonal, in which case they are equal to 1. We sometimes write I instead of In. If A = (aij) is a square matrix, we define in general its diagonal components to be the elements a jj' We have a natural ring-homomorphism of R into the ring of n x n matrices, given by

Thus cl; is the square n x n matrix having all its components equal to 0 except the diagonal components, which are equal to c. Let us denote t