Covering Numbers in Linear Algebra

We compute the minimal cardinalities of coverings and irredundant coverings of a vector space over an arbitrary field by proper linear subspaces. Analogues for affine linear subspaces are also given. Notation: The cardinality of a set S will be denoted by #S. For a vector space V over a field K , we denote its dimension by dim K . 1. LINEAR COVERINGS. Let V be a vector space over a field K . A linear covering of V is a collection {Wi }i∈I of proper K -subspaces such that V = ⋃ i∈I Wi . A linear covering is irredundant if for all J ( I , ⋃ i∈J Wi 6= V . Linear coverings exist if and only if dim V ≥ 2. The linear covering number LC(V ) of a vector space V of dimension at least 2 is the least cardinality #I of a linear covering {Wi }i∈I of V . The irredundant linear covering number ILC(V ) is the least cardinality of an irredundant linear covering of V . Thus LC(V ) ≤ ILC(V ). The main result of this note is a computation of LC(V ) and ILC(V ). Main Theorem. Let V be a vector space over a field K , with dim V ≥ 2. (a) If dim V and #K are not both infinite, then LC(V ) = #K + 1. (b) If dim V and #K are both infinite, then LC(V ) = א0. (c) In all cases we have ILC(V ) = #K + 1. Here is a counterintuitive consequence: the vector space R[t] of polynomials has a countably infinite linear covering—indeed, for each n ∈ Z, let Wn be the subspace of polynomials of degree at most n. However any irredundant linear covering of R[t] has cardinality #R+ 1 = 20 . Redundant coverings can be much more efficient! That a finite-dimensional vector space over an infinite field cannot be a finite union of proper linear subspaces is mathematical folkore: the problem and its solution appear many times in the literature. For instance problem 10707 in this MONTHLY is intermediate between this fact and our main result. The editorial comments given on page 951 of the December 2000 issue of the MONTHLY give references to variants dating back to 1959. Like many pieces of folklore, there seems to a be mild stigma against putting it in standard texts; an exception is [2, Thm. 1.2]. There are two essentially different arguments that establish this fact. Upon examination, each of these yields a stronger result, recorded as Theorem 4 and Theorem 5 below. From these two results the Main Theorem follows easily. The first two parts of the Main Theorem have appeared in the literature before (but only very recently!): they were shown by A. Khare [1]. I found these results independently in the summer of 2008. The computation of the irredundant linear covering number appears to be new. The proof of the Main Theorem is given in §2. Analogues for coverings of a vector space by affine linear subspaces appear in §3. http://dx.doi.org/10.4169/amer.math.monthly.119.01.065 MSC: Primary 15A03, Secondary 05B40 January 2012] NOTES 65 2. PROOF OF THE MAIN THEOREM. First we prove three lemmas, all consequences or special cases of the Main Theorem. Lemma 1 (Quotient Principle). Let V and W be vector spaces over a field K with dim V ≥ dim W ≥ 2. Then LC(V ) ≤ LC(W ) and ILC(V ) ≤ ILC(W ). Proof. By standard linear algebra, the hypothesis implies that there is a surjective linear map q : V → W . If {Wi }i∈I is a linear covering of W , then the complete preimages {q(Wi )}i∈I give a linear covering of V . The preimage of an irredundant covering is easily seen to be irredundant. Lemma 2. For any field K , the unique linear covering of K 2 is the set of all lines through the origin, of cardinality #K + 1. It is an irredundant covering. Proof. The set of lines through the origin is a linear covering of K 2. Moreover, any nonzero v ∈ K 2 lies on a unique line, so all lines are needed. The lines through the origin are {y = αx | α ∈ K } and x = 0, so there are #K + 1 of them. Lemma 3. Let V be a vector space over a field K , with dim V ≥ 2. Then there are at least #K + 1 hyperplanes—i.e., codimension-one linear subspaces—in V . Proof. When dim V = 2 there are exactly #K + 1 hyperplanes {L i }. In general, take a surjective linear map q : V → K 2; then {q−1(L i )} is a family of distinct hyperplanes in V . The exact number of hyperplanes in V is of course known, but not needed here. Theorem 4. Let V be a finite-dimensional vector space over a field K , and let {Wi }i∈I be a linear covering of V . Then #I ≥ #K + 1. Proof. Since every proper subspace is contained in a hyperplane, it suffices to consider hyperplane coverings. We go by induction on d , the case d = 2 being Lemma 2. Assume the result for (d − 1)-dimensional spaces and, seeking a contradiction, that we have a linear covering {Wi }i∈I of K d with #I < #K + 1. By Lemma 3, there is a hyperplane W such that W 6= Wi for any i ∈ I . Then {Wi ∩ W }i∈I is a covering of W ∼= K d−1 by at most #I < #K + 1 hyperplanes, giving a contradiction. Theorem 5. Let V be a vector space over a field K , and let {Wi }i∈I be an irredundant linear covering of V . Then #I ≥ #K + 1. Proof. Let {Wi }i∈I be an irredundant linear covering of V . Choose one of the subspaces in the covering, say W•. By irredundancy, there exists u ∈ W• \ ⋃ i 6=•Wi ; certainly there exists v ∈ V \W•. Consider the affine line ` = {tu + v | t ∈ K }; evidently #` = #K . If w = tu + v ∈ ` ∩ W•, then v = w − tu ∈ W•, giving a contradiction. Further, if for any i 6= • we had #(` ∩ Wi ) ≥ 2, then we would have ` ⊂ Wi and thus also the K -span of ` is contained in Wi , so u = (u + v) − v ∈ Wi , again giving a contradiction. It follows that #` = #K ≤ #(I \ {•}). Proof of the Main Theorem. Let V be a K -vector space of dimension at least 2. By Theorem 5, ILC(V ) ≥ #K + 1, and by Lemmas 1 and 2, ILC(V ) ≤ #K + 1. So ILC(V ) = #K + 1, proving part (c) of the Main Theorem. It remains to compute LC(V ). 66 c © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 119 Case 1. Suppose 2 ≤ dim V < א0. By Theorem 4 we have LC(V ) ≥ #K + 1, whereas by Lemma 1 and Lemma 2 we have LC(V ) ≤ LC(K 2) = #K + 1. Case 2. Suppose dim V ≥ א0 and K is finite. Then LC(V ) ≤ LC(K 2) = #K + 1 < א0. Suppose that V had a linear covering {Wi } i=1 with n < #K + 1. Then, since n is finite, we may obtain an irredundant subcovering simply by removing redundant subspaces one at a time, until we get an irredundant covering by m subspaces, with m ≤ n < #K + 1 subspaces, contradicting Theorem 5. Case 3. Suppose dim V and #K are both infinite. Consider W = ⊕ ∞ i=1 K , a vector space of dimension א0. For n ∈ Z, put Wn := ⊕n i=1 K . Then {Wn} ∞ n=1 gives a covering of W of cardinality א0. Since dim V ≥ dim W , by Lemma 1 LC(V ) ≤ א0. If V were to admit a finite linear covering, then as above it would admit a finite irredundant linear covering, contradicting Theorem 5. 3. AFFINE COVERING NUMBERS. An affine covering {Ai }i∈I of a vector space V is a covering by translates of proper linear subspaces. An affine covering is irredundant if no proper subset gives a covering. The affine covering number AC(V ) is the least cardinality of an affine covering, and similarly the irredundant affine covering number IAC(V ) is the least cardinality of an irredundant affine covering. Theorem 6. Let V be a vector space over a field K , with dim V ≥ 1. (a) If min(dim V, #K ) is finite, then AC(V ) = #K . (b) If dim V and #K are both infinite, then AC(V ) = א0. (c) We have IAC(V ) = #K . The proof of the Main Theorem goes through with minor modifications. Lemma 1 holds verbatim. The following self-evident result is the analogue of Lemma 2. Lemma 7. For any field K , the unique affine covering of K 1 is the set of all points of K , of cardinality #K . It is an irredundant covering. Combining these two results we get the analogue of Lemma 3, in which #K + 1 is replaced by #K . To prove the analogue of Theorem 4, note that for two codimensionone affine subspaces W1, W2 of a vector space V , W1 ∩ W2 is either empty or is a codimension-one affine subspace in each Wi . In the proof of the analogue of Theorem 5 we use the line ` = {(1− t)u + tv | t ∈ K }. ACKNOWLEDGMENTS. This work was partially supported by National Science Foundation grant DMS-0701771. REFERENCES1. Khare, A., Vector spaces as unions of proper subspaces, Linear Algebra Appl. 431 (2009) 1681–1686,available at http://dx.doi.org/10.1016/j.laa.2009.06.001.2. Roman, S., Advanced linear algebra, third edition, Graduate Texts in Mathematics, vol. 135, Springer-Verlag, New York, 2008. Department of Mathematics, University of Georgia, Athens, GA [email protected] January 2012]NOTES67