Dual bases

One more word before embarking on the proofs of the important theorems. The concept of dual space was defined without any reference to coordinate systems; a glance at the following proofs will show a superabundance of coordinate systems. We wish to point out that this phenomenon is inevitable; we shall be establishing results concerning dimension, and dimension is the one concept (so far) whose very definition is given in terms of a basis.

Theorem 1. If is an n-dimensional vector space, if is a basis in , and if is any set of scalars, then there is one and only one linear functional on such that for .

Proof. Every in may be written in the form in one and only one way; if is any linear functional, then From this relation the uniqueness of is clear; if , then the value of is determined, for every , by . The argument can also be turned around; if we define by then is indeed a linear functional, and . ◻

Theorem 2. If is an -dimensional vector space and if is a basis in , then there is a uniquely determined basis in , , with the property that . Consequently the dual space of an -dimensional space is -dimensional.

The basis is called the dual basis of .

Proof. It follows from Theorem 1 that, for each , a unique in can be found so that ; we have only to prove that the set is a basis in .

In the first place, is a linearly independent set, for if we had , in other words, if for all , then we should have, for ,

In the second place, every in is a linear combination of . To prove this, write ; then, for , we have

On the other hand

so that, substituting in the preceding equation, we get

Consequently , and the proof of the theorem is complete. ◻

We shall need also the following easy consequence of Theorem 2.

Theorem 3. If and are any two different vectors of the -dimensional vector space , then there exists a linear functional on such that ; or equivalently, to any non-zero vector in there corresponds a in such that .

Proof. That the two statements in the theorem are indeed equivalent is seen by considering . We shall, accordingly, prove the latter statement only.

Let be any basis in , and let be the dual basis in . If , then (as above) . Hence if for all , and, in particular, if for , then . ◻

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