Leigh Stewart

29 April, 1999

Journal Problem

Math 232A

 

Unions and Intersections of Subspaces

 

The Greeks of antiquity knew of only one space and one geometry. To them space was an absolute concept—a realm or central point in which objects are able to be moved about with freedom. The seventeenth century ushered in the ideas of analytic geometry, and space was redefined as a collection of points. The nineteenth century saw the emergence of new, non-Euclidean geometries; mathematicians came to believe that there was more than one geometry, but they still conceived space as a collection of points within a locus which could be compared with one another. In the early twentieth century the study of abstract spaces dawned. Space was no longer defined by the tidy classifications of earlier centuries; it was redefined more generally as "merely a set of points and a set of relations in which those points are involved" (Eves, p463). One special classification of spaces are linear subspaces, which are the topic of this paper. A subspace is defined as any subset of Rn which contains the zero vector, is closed under addition, and is closed under scalar multiplication (Bretscher, p146). Understanding subspaces offers a good basis to understanding the abstract spaces which have redefined the way mathematicians think about geometry and space.

 

In this paper we will begin with the premise V and W are two subspaces of Rn and ask two questions. Question1: Is the intersection VÇ W necessarily a subspace? Question2: For which cases of V and W is VÈ W necessarily a subspace? From these questions we will form two conjectures, respectively. Conjecture1 states that VÇ W is always a subspace when V and W are subspaces of Rn. Conjecture2 is that for any subspaces V and W, VÈ W is a subspace if and only if VÌ W or WÌ V. We will then prove the conjectures to create helpful theorems about the characteristics of the intersection and union of two subspaces.

 

Question1: Given two subspaces of Rn, V and W, is VÇ W necessarily a subspace?

From this motivating question we form conjecture1: VÇ W is always a subspace when V and W are subspaces of Rn. We know that for any subset to be a subspace, it must meet the following three conditions:

  1. The subset must contain the zero vector which is an element of Rn.
  2. The subset must be closed under addition such that if vectors x and y are both contained in the subset then so is x+y.
  3. The subset must be closed under scalar multiplication such that if the vector x is in the subset and c is an arbitrary scalar then cx is also in the subset. (Bretscher, p146).

To prove our first conjecture we must simply prove that conditions i, ii and iii hold for all VÇ W where V and W are subspaces of Rn.

Proof:

  1. Does VÇ W contain the zero vector? Yes.

We know V contains the zero vector and we know W contains the zero vector. Since it is common to both sets it must be in the set that is the intersection of the subspaces.

  1. Is VÇ W closed under addition? Yes.

Pick any vectors x1 and x2 which are elements of the set VÇ W. We know that x1+ x2 is necessarily in VÇ W because both are in V and since V is a subspace x1+ x2 must be contained in V. Similarly both vectors are in W and since W is a subspace x1+ x2 must be in W. Since x1+ x2 is common to both V and W it must be contained in VÇ W as well.

  1. Is VÇ W closed under scalar multiplication? Yes.

Given any vector x in VÇ W and any arbitrary scalar c, cx is in V because V is a subspace and cx is in W because W is a subspace. Since cx is common to both V and W it must be contained in VÇ W as well.

Since VÇ W satisfies the three aforementioned requirements we have proven it to be a subspace when V and W are subspaces. We have proven our first theorem.

Theorem1: VÇ W is always a subspace when V and W are subspaces of Rn.

Now that we have drawn conclusions about VÇ W, let us look at VÈ W.

Question2: For which cases of V and W is VÈ W necessarily a subspace?

Drawing on this question, we will work with conjecture2: For any subspaces V and W, VÈ W is a subspace if and only if WÌ V. Since our second conjecture is an "if and only if" statement, we must prove it holds in both directions. First we will prove that if it is the case that WÌ V, VÈ W is necessarily a subspace. Second we will prove that if VÈ W is a subspace, WÌ V must hold.

Proof:

(1) Let us first look at the case if WÌ V, then VÈ W is necessarily a subspace. If V and W are both subspaces, WÌ V implies that VÈ W=V. It is already given that V is a subspace, so we see that VÈ W is a subspace as well.

(2) Let us now show that when V and W are subspaces of Rn, WÌ V if VÈ W is also a subspace Rn. When combining two subspaces to make a third, we run into problems only when the union of the subspaces is not closed under addition. Therefore we will prove that if VÈ W is a subspace, it must be closed under addition and therefore V and W much have a relationship that allows for this closure. That relationship, we will show, is that WÌ V.

If VÈ W is a subspace, let us suppose VË W. If this is the case then there is some v0 contained in V but not contained in W. We can note that for any w contained in W, v0+w is contained in VÈ W. This implies one of two cases: either v0+w is contained in W or v0+w is contained in V.

  1. The first case, that v0+w is contained in W implies that v0 is an element of W, which we know from above is false.
  2. The second case, that v0+w is contained in V implies that w is an element of V for all w.

Since (i) is a false statement and (ii) must therefore be the case that holds, we have shown that if V Ë W, then WÌ V. So we see that if V, W and VÈ W are all subspaces, WÌ V must hold.

Thus we have proven both requirements of our "if and only if" statement, hence proving conjecture2 from which we may state:

Theorem2: For any subspaces V and W, VÈ W is a subspace if and only if WÌ V.

[Editor's Note: The proof actually shows that VÈ W is a subspace if and only if WÌ V OR VÌ W.]

 

We have proven the two initially proposed conjectures, and they can now be considered theorems. These theorems reveal the behavior of subspaces when they are combined with one another via intersection or union. It is important to understand subspaces and the rules that govern their relationships with one another because subspaces of Rn form part of the conceptual foundation necessary to study abstract linear spaces.

 

 

 

 

Works Cited

 

Bretscher, Otto. Linear Algebra With Applications. New Jersey: Prentice Hall, 1997.

 

Eves, Howard. An Introduction to the History of Mathematics. 5th ed. Philadelphia: CBS College Publishing, 1983.