Projections

Especially important for our purposes is another connection between direct sums and linear transformations.

Definition 1. If is the direct sum of and , so that every in may be written, uniquely, in the form with in and , the projection on along is the transformation defined by .

If direct sums are important, then projections are also, since, as we shall see, they are a very powerful algebraic tool in studying the geometric concept of direct sum. The reader will easily satisfy himself about the reason for the word "projection" by drawing a pair of axes (linear manifolds) in the plane (their direct sum). To make the picture look general enough, do not draw perpendicular axes!

We skipped over one point whose proof is easy enough to skip over, but whose existence should be recognized; it must be shown that is a linear transformation. We leave this verification to the reader, and go on to look for special properties of projections.

Theorem 1. A linear transformation is a projection on some subspace if and only if it is idempotent, that is, .

Proof. If is the projection on along , and if , with in and in , then the decomposition of is , so that Conversely, suppose that . Let be the set of all vectors in for which ; let be the set of all vectors for which . It is clear that both and are subspaces; we shall prove that . In view of the theorem of Section: Direct sums , we need to prove that and are disjoint and that together they span .

If is in , then ; if is in , then ; hence if is in both and , then . For an arbitrary we have If we write and , then and so that is in and is in . This proves that , and that the projection on along is precisely . ◻

As an immediate consequence of the above proof we obtain also the following result.

Theorem 2. If is the projection on along , then and are, respectively, the sets of all solutions of the equations and .

By means of these two theorems we can remove the apparent asymmetry, in the definition of projections, between the roles played by and . If to every we make correspond not but , we also get an idempotent linear transformation. This transformation (namely, ) is the projection on along . We sum up the facts as follows.

Theorem 3. A linear transformation is a projection if and only if is a projection; if is the projection on along , then is the projection on along .

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