Vol. 10 No 5 - Pi Mu Epsilon

VIAR, DIMENSION FORMULA 377subspace and applying the case n = 2 we havedim(Vl + - +Vn) = dimVn + dim(V1 + - +Vn.,) - dim(Vnn(Vl + - +V,,.,))3 78 PI MU EPSILON JOURNALProof. This is really a corollary to the first theorem. Take dimensions of bothsides. The dimension of the left side is the same as the dimension of the right sideby the first theorem. Since the left side and the right side have the same dimensionand they are over the same field they must be isomorphic.The second theorem isn't very satisfying. It gives us a way of looking atthe dimension formula of the first theorem in terms of quotient spaces, but thegiven isomorphism is "unnatural" in that we haven't given the isomorphism andit seems unlikely that one will be found. Is there a better generalization of thedimension formula, and does it lead to a natural generalization of the secondisomorphism theorem? We leave this question the reader.Hence the theorem follows by induction. DThe problem with our formula is that it is not very symmetric inV,, V,, - , Vn . To get a more symmetric formula, write down our formula n!times (once for each permutation of {Vl, V,, - , Vn}), and then add each column.For example, in the case n = 3 this leads to-2dim(VlnV3)-2dim(V2nV3)- 2dim(Vl n(V2 +V3))- 2dim(V2n(V, +VJ)- 2dim(V3n(Vl +VJ)We leave it to the reader to write down a general form for this.Recall that the second isomorphism theorem for vector spaces says thatif V, and V, are subspaces thenTaking dimensions of both sides of this formula yields the well known formulathat we started with. One would guess that our dimension formula must also giverise to a quotient isomorphism. In fact, we have the following.Theorem 2. If Vl, V,,-, Vn are subspaces of a finite dimensional vector space,then