Calculus of subspaces

Theorem 1. The intersection of any collection of subspaces is a subspace.

Proof. If we use an index to tell apart the members of the collection, so that the given subspaces are , let us write Since every , contains , so does , and therefore is not empty. If and belong to (that is, to all ), then belongs to all , and therefore is a subspace. ◻

To see an application of this theorem, suppose that is an arbitrary set of vectors (not necessarily a subspace) in a vector space . There certainly exist subspaces containing every element of (that is, such that ); the whole space is, for example, such a subspace. Let be the intersection of all the subspaces containing ; it is clear that itself is a subspace containing . It is clear, moreover, that is the smallest such subspace; if is also contained in the subspace , , then . The subspace so defined is called the subspace spanned by or the span of . The following result establishes the connection between the notion of spanning and the concepts studied in Sections 5–9.

Theorem 2. If is any set of vectors in a vector space and if is the subspace spanned by , then is the same as the set of all linear combinations of elements of .

Proof. It is clear that a linear combination of linear combinations of elements of may again be written as a linear combination of elements of . Hence the set of all linear combinations of elements of is a subspace containing ; it follows that this subspace must also contain . Now turn the argument around: contains and is a subspace; hence contains all linear combinations of elements of . ◻

We see therefore that in our new terminology we may define a linear basis as a set of linearly independent vectors that spans the whole space.

Our next result is an easy consequence of Theorem 2; its proof may be safely left to the reader.

Theorem 3. If and are any two subspaces and if is the subspace spanned by and together, then is the same as the set of all vectors of the form , with in and in .

Prompted by this theorem, we shall use the notation for the subspace spanned by and . We shall say that a subspace of a vector space is a complement of a subspace if and .

Quick Links

Social Media