Page 11 - A Course in Linear Algebra with Applications

X Preface

position of matrices in the entire theory makes this a logical
and reasonable course. However the motivation for the in-
troduction of matrices, by means of linear equations, is still
provided informally. The second chapter forms a basis for
the whole subject with a full account of the theory of linear
equations. This is followed by a chapter on determinants, a
topic that has been unfairly neglected recently. In practice it
is hard to give a satisfactory definition of the general n x n
determinant without using permutations, so a brief account
of these is given.
Chapters Five and Six introduce the student to vector
spaces. The concept of an abstract vector space is probably
the most challenging one in the entire subject for the non-
mathematician, but it is a concept which is well worth the
effort of mastering. Our approach proceeds in gentle stages,
through a series of examples that exhibit the essential fea-
tures of a vector space; only then are the details of the def-
inition written down. However I feel that nothing is gained
by ducking the issue and omitting the definition entirely, as is
sometimes done.
Linear tranformations are the subject of Chapter Six.
After a brief introduction to functional notation, and numer-
ous examples of linear transformations, a thorough account
of the relation between linear transformations and matrices is
given. In addition both kernel and image are introduced and
are related to the null and column spaces of a matrix.
Orthogonality, perhaps the heart of the subject, receives
an extended treatment in Chapter Seven. After a gentle in-
troduction by way of scalar products in three dimensions —
which will be familiar to the student from calculus — inner
product spaces are denned and the Gram-Schmidt procedure
is described. The chapter concludes with a detailed account
of The Method of Least Squares, including the problem of