HIGHER ALGEBRA Much that is elementary in the sense of elemental, is not so in the sense of being adapted to the pre-university schools. Higher algebra of a sort is taught in these schools, and higher algebra of an entirely different sort is taught in the universities. In the pre-university schools the work serves as an introduction to the theory of equations, algebraic numbers, determinants, certain series, inequalities, limits and logarithms, and the simpler fea tures of analytic trigonometry, analytic geometry and the differen tial calculus. In the universities it includes work in the theory of equations, the study of polynomials, combinatorial analysis, prob ability, matrices and various other fields. For the more important of these topics see AGGREGATES ; ANALYSIS; ANALYTIC GEOMETRY ; BARYCENTRIC CALCULUS ; BINOMIAL FORMULA; TENSOR ANALY SIS; CALCULUS OF DIFFERENCES ; COMBINATORIAL ANALYSIS; COMPLEX NUMBERS ; DETERMINANT ; EQUATIONS ; DIFFERENTIAL FORMS. ALGEBRAIC FORMS; GRAPHIC METHODS ; GROUPS; For one set of variables it is often useful to substitute expres sions containing a second, different set. An example of this is the change of axes, or planes, of reference in analytic geometry. Those expressions are most often linear (i.e., of the order one) in the new system, and the change is called a linear transformation. For some reason, however, expressions of the second or higher order may be used; this case is less frequent and less simple, and can be treated as a linear transformation in an increased number of variables, plus the inclusion of an additional locus in the sys tem under discussion. By any linear transformation, if the new variables are given the same names as the old, the old form is changed into a new one. Two forms so derivable one from the other are called equivalent under the transformation. By this defi nition, any form is equivalent to all that can be derived from it by linear transformations. Consider a binary quadric:— and F(x) is equivalent to f(x). Such a transformation may be represented, between two binary forms, by means of indeter minate coefficients or parameters ai , ce2 l31 02 thus:— and it is reversible without ambiguity if the determinant (aii32—«201) is not zero, for then these two equations can be solved for yi and y2.
One reason for the employment of linear transforma tions is that aLl such form a group. (See GRouPs.) That is, if two linear transformations are made in succession, variables (x) into functions of (y), and the (y) into functions of variables (z), then the (x) are linear functions of the (z). The linearity is un changed by compounding two or more linear transformations. One speaks often of equivalence under the linear group, meaning to exclude transformations whose determinant is zero; then the equivalence is a mutual relation. There are groups of linear transformations that do not include all such substitutions; just as there are in geometry rotations of a plane about one fixed centre, leaving a system of concentric circles unaltered. For some purposes, the group of linear substitutions whose deter minant is I are applied; or the group of real transformations, ex cluding all that have any imaginary parameters. If imaginary sub stitutions are admitted, a quaternary quadric
x32 x42 is equivalent to another xi2+ x22 — x32 — x42; not, however, if the group is that of real linear transformations. Two systems of forms in any number of sets of variables are equivalent under the group if the members of one system can be transformed into those of the other system by means of separate linear substitutions on all the different sets of variables involved. Two sets of variables that are subject to the same transformation are cogredient.
What happens to the coeffi cients of the various terms in a form, or quantic, when it is transformed into an equivalent quantic? It is seen, on trial, that the new coefficients contain the old in the first degree, but the, parameters of the transformation to higher degrees, each form to a degree equal to its order in the variables. Under linear trans formation of its variables, every form undergoes also a reverse linear transformation of its coefficients, determined or induced by that of the variables. These need not be described more minutely here; they require for precise statement the definition of polars. (See POLE and POLAR.) While the two forms are called equivalent when the equations that transform variables are given, to make the statement algebraically exact the coefficients of similar terms in the two forms must be equated, and these equa tions constitute the induced substitution. But two forms may be given with numerical coefficients. By what means may their equivalence be tested? This is the first major problem in alge braic form theory. The answer is far from complete. Sufficient conditions can be stated, but many of them are certainly re dundant. The minimum set of necessary conditions is known in only a few particular cases. To find explicitly the parameters of a linear substitution from the equations of the induced sub stitution on the coefficients would require the solving of equa tions whose degree in the parameters is equal to that of the given forms in the proper variables. That calls for extraction of roots or other irrational operations. Naturally an answer is preferred which should demand only rational operations in proving equiva lence or non-equivalence. Hence comes a re-formulation of the question: When two forms (or systems of forms) are equivalent, what rational functions of the coefficients, or coefficients and variables, will have the same numerical value for both forms? What rational functions of coefficients and variables retain the same value, remain invariant, under the group of linear substitu tions? This question can be answered. As an exercise, the reader might verify that two binary cubics are generally equivalent, but not always two binary quartics, taken at random.
An invariant or covariant (a comitant) of a given form or system is any function of the co efficients, or coefficients and variables, which is identically equal to the same function of the corresponding coefficients and vari ables in every equivalent form or system—equal, that is, save for a multiplicative constant depending on the parameters of the transformation. Such a constant, it can be proved, is always some power of the determinant of the substitution. These are relative invariants, but if the multiplicative constant is unity, the invariant is absolute.
