We are now ready to proceed with multilinear algebra. The basic concept is that of multilinear form (or functional), an easy generalization of the concept of bilinear form. Suppose that \(\mathcal{V}_1, \ldots, \mathcal{V}_k\) are vector spaces (over the same field); a k-linear form ( \(k = 1, 2, 3, \ldots\) ) is a scalar-valued function on the direct sum \(\mathcal{V}_1 \oplus \cdots \oplus \mathcal{V}_k\) with the property that for each fixed value of any \(k-1\) arguments it depends linearly on the remaining argument. The \(1\) -linear forms are simply the linear functionals (on \(\mathcal{V}_1\) ), and the \(2\) -linear forms are the bilinear forms (on \(\mathcal{V}_1 \oplus \mathcal{V}_2\) ). The \(3\) -linear (or trilinear) forms are the scalar-valued functions \(w\) (on \(\mathcal{V}_1 \oplus \mathcal{V}_2 \oplus \mathcal{V}_3\) ) such that \[w(\alpha_1 x_1 + \alpha_2 x_2, y, z) = \alpha_1 w(x_1, y, z) + \alpha_2 w(x_2, y, z),\] and such that similar identities hold for \(w(x, \alpha_1 y_1 + \alpha_2 y_2, z)\) and \(w(x, y, \alpha_1 z_1 + \alpha_2 z_2)\) . A function that is \(k\) -linear for some \(k\) is called a multilinear form .

Much of the theory of bilinear forms extends easily to the multilinear case. Thus, for instance, if \(w_1\) and \(w_2\) are \(k\) -linear forms, if \(\alpha_1\) and \(\alpha_2\) are scalars, and if \(w\) is defined by \[w(x_1, \ldots, x_k) = \alpha_1 w_1(x_1, \ldots, x_k) + \alpha_2 w_2(x_1, \ldots, x_k)\] whenever \(x_i\) is in \(\mathcal{V}_i\) , \(i = 1, \ldots, k\) , then \(w\) is a \(k\) -linear form, denoted by \(\alpha_1 w_1 + \alpha_2 w_2\) . The set of all \(k\) -linear forms is a vector space with respect to this definition of the linear operations; the dimension of that vector space is the product \(n_1 \ldots n_k\) , where, of course, \(n_i\) is the dimension of \(\mathcal{V}_i\) . The proofs of all these statements are just like the proofs (in Section: Bilinear forms ) of the corresponding statements for the bilinear case. We could go on imitating the bilinear theory and, in particular, studying multiple tensor products. In order to hold our multilinear digression to a minimum, we shall proceed instead in a different, more special, and, for our purposes, more useful direction.

In what follows we shall restrict our attention to the case in which the \(k\) spaces \(\mathcal{V}_i\) are all equal to one and the same vector space, say, \(\mathcal{V}\) ; we shall assume that \(\mathcal{V}\) is finite-dimensional. In this case we shall call a “ \(k\) -linear form on \(\mathcal{V}_1 \oplus \cdots \oplus \mathcal{V}_k\) ” simply a “ \(k\) -linear form on \(\mathcal{V}\) ,” or, even more simply, a “ \(k\) -linear form”; the language is slightly inaccurate but, in context, completely unambiguous. If the dimension of \(\mathcal{V}\) is \(n\) , then the dimension of the vector space of all \(k\) -linear forms is \(n^k\) . The space \(\mathcal{V}\) and, of course, the dimension \(n\) will be held fixed throughout the following discussion.

The special character of the case we are studying enables us to apply a technique that is not universally available; the technique is to operate on \(k\) -linear forms by permutations in \(\mathcal{S}_k\) . If \(w\) is a \(k\) -linear form, and if \(\pi\) is in \(\mathcal{S}_k\) , we write \[\pi w(x_1, \ldots, x_k) = w(x_{\pi(1)}, \ldots, x_{\pi(k)})\] whenever \(x_1, \ldots, x_k\) are in \(\mathcal{V}\) . The function \(\pi w\) so defined is again a \(k\) -linear form. (The value of \(\pi w\) at \((x_1, \ldots, x_k)\) is more honestly denoted by \((\pi w)(x_1, \ldots, x_k)\) ; since, however, the simpler notation does not appear to lead to any confusion, we shall continue to use it.)

Using the way permutations act on \(k\) -linear forms, we can define some interesting sets of such forms. Thus, for instance, a \(k\) -linear form \(w\) is called symmetric if \(\pi w = w\) for every permutation \(\pi\) in \(\mathcal{S}_k\) . (Note that if \(k = 1\) , then this condition is trivially satisfied.) The set of all symmetric \(k\) -linear forms is a subspace of the space of all \(k\) -linear forms. Hence, in particular, the origin of that space, the \(k\) -linear form \(0\) , is symmetric. For a non-trivial example, suppose that \(k = 2\) , let \(y_1\) and \(y_2\) be linear functionals on \(\mathcal{V}\) , and write \[w(x_1, x_2) = y_1(x_1)y_2(x_2) + y_1(x_2)y_2(x_1).\] This procedure for constructing \(k\) -linear forms has useful generalizations. Thus, for instance, if \(1 \leq h < k \leq n\) , and if \(u\) is an \(h\) -linear form and \(v\) is a \((k - h)\) -linear form, then the equation \[w(x_1, \ldots, x_k) = u(x_1, \ldots, x_h) \cdot v(x_{h+1}, \ldots, x_k)\] defines a \(k\) -linear form \(w\) , which, in general, is not symmetric. A symmetric \(k\) -linear form can be obtained from \(w\) (or, for that matter, from any given \(k\) -linear form) by forming \(\sum \pi w\) , where the summation is extended over all permutations \(\pi\) in \(\mathcal{S}_k\) .

We shall not study symmetric \(k\) -linear forms any more. We introduced them here because they constitute a very natural class of functions definable in terms of permutations. We abandon them now in favor of another class of functions, which play a much greater role in the theory.