We know already that if \(\mathcal{M}\) is a subspace of a vector space \(\mathcal{V}\) , then there are, usually, many other subspaces \(\mathcal{N}\) in \(\mathcal{V}\) such that \(\mathcal{M} \oplus \mathcal{N} = \mathcal{V}\) . There is no natural way of choosing one from among the wealth of complements of \(\mathcal{M}\) . There is, however, a natural construction that associates with \(\mathcal{M}\) and \(\mathcal{V}\) a new vector space that, for all practical purposes, plays the role of a complement of \(\mathcal{M}\) . The theoretical advantage that the construction has over the formation of an arbitrary complement is precisely its “natural” character, i.e., the fact that it does not depend on choosing a basis, or, for that matter, on choosing anything at all.

In order to understand the construction it is a good idea to keep a picture in mind. Suppose, for instance, that \(\mathcal{V} = \mathbb{R}^2\) (the real coordinate plane) and that \(\mathcal{M}\) consists of all those vectors \((\xi_1, \xi_2)\) for which \(\xi_2 = 0\) (the horizontal axis). Each complement of \(\mathcal{M}\) is a line (other than the horizontal axis) through the origin. Observe that each such complement has the property that it intersects every horizontal line in exactly one point. The idea of the construction we shall describe is to make a vector space out of the set of all horizontal lines.

We begin by using \(\mathcal{M}\) to single out certain subsets of \(\mathcal{V}\) . (We are back in the general case now.) If \(x\) is an arbitrary vector in \(\mathcal{V}\) , we write \(x + \mathcal{M}\) for the set of all sums \(x + y\) with \(y\) in \(\mathcal{M}\) ; each set of the form \(x + \mathcal{M}\) is called a coset of \(\mathcal{M}\) . (In the case of the plane-line example above, the cosets are the horizontal lines.) Note that one and the same coset can arise from two different vectors, i.e., that even if \(x \ne y\) , it is possible that \(x + \mathcal{M} = y + \mathcal{M}\) . It makes good sense, just the same, to speak of a coset, say \(\mathcal{H}\) , of \(\mathcal{M}\) , without specifying which element (or elements) \(\mathcal{H}\) comes from; to say that \(\mathcal{H}\) is a coset (of \(\mathcal{M}\) ) means simply that there is at least one \(x\) such that \(\mathcal{H} = x + \mathcal{M}\) .

If \(\mathcal{H}\) and \(\mathcal{K}\) are cosets (of \(\mathcal{M}\) ), we write \(\mathcal{H} + \mathcal{K}\) for the set of all sums \(u + v\) with \(u\) in \(\mathcal{H}\) and \(v\) in \(\mathcal{K}\) ; we assert that \(\mathcal{H} + \mathcal{K}\) is also a coset of \(\mathcal{M}\) . Indeed, if \(\mathcal{H} = x + \mathcal{M}\) and \(\mathcal{K} = y + \mathcal{M}\) , then every element of \(\mathcal{H} + \mathcal{K}\) belongs to the coset \((x + y) + \mathcal{M}\) (note that \(\mathcal{M} + \mathcal{M} = \mathcal{M}\) ), and, conversely, every element of \((x + y) + \mathcal{M}\) is in \(\mathcal{H} + \mathcal{K}\) . (If, for instance, \(z\) is in \(\mathcal{M}\) , then \((x + y) + z = (x + z) + (y + 0)\) .) In other words, \(\mathcal{H} + \mathcal{K} = (x + y) + \mathcal{M}\) , so that \(\mathcal{H} + \mathcal{K}\) is a coset, as asserted. We leave to the reader the verification that coset addition is commutative and associative. The coset \(\mathcal{M}\) (i.e., \(0 + \mathcal{M}\) ) is such that \(\mathcal{H} + \mathcal{M} = \mathcal{H}\) for every coset \(\mathcal{H}\) , and, moreover, it is the only coset with this property. (If \((x + \mathcal{M}) + (y + \mathcal{M}) = x + \mathcal{M}\) , then \(x + \mathcal{M}\) contains \(x + y\) , so that \(x + y = x + u\) for some \(u\) in \(\mathcal{M}\) ; this implies that \(y\) is in \(\mathcal{M}\) , and hence that \(y + \mathcal{M} = \mathcal{M}\) .) If \(\mathcal{H}\) is a coset, then the set consisting of all those vectors \(-u\) , with \(u\) in \(\mathcal{H}\) , is itself a coset, which we shall denote by \(-\mathcal{H}\) . The coset \(-\mathcal{H}\) is such that \(\mathcal{H} + (-\mathcal{H}) = \mathcal{M}\) , and, moreover, \(-\mathcal{H}\) is the only coset with this property. To sum up: the addition of cosets satisfies the axioms (A) of Section: Vector spaces .

If \(\mathcal{H}\) is a coset and if \(\alpha\) is a scalar, we write \(\alpha \mathcal{H}\) for the set consisting of all the vectors \(\alpha u\) with \(u\) in \(\mathcal{H}\) in case \(\alpha \ne 0\) ; the coset \(0 \cdot \mathcal{H}\) is defined to be \(\mathcal{M}\) . A simple verification shows that this concept of multiplication satisfies the axioms (B) and (C) of Section: Vector spaces .

The set of all cosets has thus been proved to be a vector space with respect to the linear operations defined above. This vector space is called the quotient space of \(\mathcal{V}\) modulo \(\mathcal{M}\) ; it is denoted by \(\mathcal{V}/\mathcal{M}\) .