and Subspaces of

We are familiar with two operations that can be applied to vectors in , namely, addition and scalar multiplication. We learned that addition and scalar multiplication satisfy many nice properties (see Theorem th:vecproperties). These properties give an algebraic structure. We begin this section by introducing another property, called closure. Adding closure to the properties we studied earlier allows us to show that satisfies all of the properties of a vector space.

Closure

as a Vector Space

In Theorem th:vecproperties we learned that vector addition and scalar multiplication in satisfy the following eight properties:

For all vectors , , , and scalars ,

(a)
Commutative Property of Addition:
(b)
Associative Property of Addition:
(c)
Existence of Additive Identity:
(d)
Existence of Additive Inverse:
(e)
Distributive Property over Vector Addition:
(f)
Distributive Property over Scalar Addition:
(g)
Associative Property for Scalar Multiplication:
(h)
Multiplication by :

In addition, observe that

  • is closed under addition (Why?)
  • is closed under scalar multiplication (Why?)

The eight properties of vector operations, together with closure, constitute the criteria for a set with two operations to be considered a vector space. So, is a vector space.

We will encounter other vector spaces later. Any vector space must be closed under both of its operations, and must satisfy the other eight properties in the list above. We will see (in Abstract Vector Spaces, for instance) that a wide variety of sets, with a wide variety of operations, are vector spaces (one important reason to study linear algebra). As we shall see, these sets and their operations may look very different, but the behavior of the elements under the two operations makes them vector spaces. For now, we simply focus on .

Subspaces of

Now that we understand what it means for a set to be closed under addition and scalar multiplication, we are ready for the main definition.

We use the term subspace because it turns out that any subset of closed under both addition and scalar multiplication is also a vector space. In other words, by inheriting vector addition and scalar multiplication from , and satisfying the properties of closure, a subset of will automatically satisfy all vector space properties. We will prove this in Theorem th:subspacetestabstract of Abstract Vector Spaces.

Recall that the span of a set of vectors is the set of all linear combinations of those vectors (see Definition def:span). It is easy to see from this definition, that the span of any set of vectors in must be closed under both addition and scalar multiplication, and therefore the span of those vectors is a subspace of . This argument proves the following result, giving us an abundance of examples of subspaces:

In particular, if we take to be the single vector , we have that is a subspace of . Geometrically, this subspace is a line in the direction of the vector . Similarly, the span of two vectors is a subspace of . If the two vectors are linearly independent, then the subspace is a plane in .

Not every line or plane in is a subspace, however. The following important result provides us with a quick way to determine that some subsets are not subspaces.

Proof
Take any vector in , and note that is in because is closed under scalar multiplication.

Theorem th:zero_in_subspace shows that the only lines in that are subspaces are those that pass through the origin. The same holds true for planes and hyperplanes. For example, the plane in is not a subspace of , while any plane containing the origin is a subspace.

The proof is similar to what was done for the previous theorem and is left as an exercise.

Practice Problems

Problems prob:Y^+1-prob:Y^+2 Let be the set of all vectors in whose components are non-negative.

Is closed under vector addition?
Yes No
Is closed under scalar multiplication?
Yes No

Problems pr:R^3axes1-pr:R^3axes2 Let be the set of all vectors in that lie on either the -axis, the -axis, or the -axis.

Is closed under vector addition?
Yes No
Is closed under scalar multiplication?
Yes No

Problems prob:closedmultchoice1-prob:closedmultchoice2 For each figure below, determine whether the set of vectors shown in the figure is closed under vector addition and scalar multiplication. Justify your responses.

consists of all vectors in in a slanted half-plane which has the -axis as a boundary.

Is closed under scalar multiplication?

Yes No

Is closed under addition?

Yes No
consists of all vectors along the line, as shown.

Is closed under scalar multiplication?

Yes No

Is closed under addition?

Yes No
Prove Theorem th:opposite_in_subspace.
Let be an matrix. Let be the subset of consisting of all vectors such that . Prove that is a subspace of . (Note that this subspace is called the null space of the matrix and we will denote it .)