By K. W. GRUENBERG

This is basically a e-book on linear algebra. however the process is a bit of strange in that we emphasise in the course of the geometric element of the topic. the cloth is acceptable for a path on linear algebra for mathe matics majors at North American Universities of their junior or senior yr and at British Universities of their moment or 3rd 12 months. in spite of the fact that, in view of the constitution of undergraduate classes within the usa, it's very attainable that, at many associations, the textual content will be stumbled on improved first and foremost graduate point. The booklet has goals: to supply a uncomplicated direction in linear algebra as much as, and together with, modules over a imperative excellent area; and to give an explanation for in rigorous language the intuitively prevalent techniques of euclidean, affine, and projective geometry and the kin among them. it truly is more and more known that linear algebra can be approached from a geometrical perspective. this is applicable not just to arithmetic majors but additionally to mathematically-oriented typical scientists and engineers.

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**Example text**

M, span M i then M is the solution space of the equations (2). There is a natural interpretation of these results for an affine or a projective geometry for which a coordinate system is given. If A is an affine geometry with a coordinate system a; and S an element of A, then by Proposition 2, Sa; is the set of all solutions of a suitable system oflinear equations (1). It is usual to say that the equations (1) are a set of equations for S in the given coordinate system. There is a similar terminology for projective geometries.

The subspace Mil. consisting of all (zv ... , z",) in F'" satisfying for all (YI' ... , Y",) in Mi clearly contains M. But Mil. has dimension n - (n - dim M) = dim M and so Mil. = M (by Theorem 2( 1) of Chapter I (p. 11». If the rows (au, ... , a l ",), i = 1, ... , m, span M i then M is the solution space of the equations (2). There is a natural interpretation of these results for an affine or a projective geometry for which a coordinate system is given. If A is an affine geometry with a coordinate system a; and S an element of A, then by Proposition 2, Sa; is the set of all solutions of a suitable system oflinear equations (1).

30 LINEAR GEOMETRY CHAP. II The relationship between affine and projective geometries is stated in Theorem 7, at the beginning of the next section. The reader is urged to look now at the statement of that theorem and in particular at part (6) since he will then appreciate the motivation for the following definition. DEFINITION. If M is a subspace of a vector space V, the projective dimension of M is defined to be dim M - 1 and will be denoted by pdim M. The dimension of the geometry &l( V), written pdim &l( V), will be pdim V.