Linear Algebra/Projection Onto a Subspace

''This subsection, like the others in this section, is optional. It also requires material from the optional earlier subsection on Combining Subspaces.''

The prior subsections project a vector onto a line by decomposing it into two parts: the part in the line $$\mbox{proj}_{[\vec{s}\,]}({\vec{v}\,})$$ and the rest $$\vec{v}-\mbox{proj}_{[\vec{s}\,]}({\vec{v}\,})$$. To generalize projection to arbitrary subspaces, we follow this idea.

This definition doesn't involve a sense of "orthogonal" so we can apply it to spaces other than subspaces of an $$\mathbb{R}^n$$. (Definitions of orthogonality for other spaces are perfectly possible, but we haven't seen any in this book.)

A natural question is: what is the relationship between the projection operation defined above, and the operation of orthogonal projection onto a line? The second picture above suggests the answer&mdash; orthogonal projection onto a line is a special case of the projection defined above; it is just projection along a subspace perpendicular to the line. In addition to pointing out that projection along a subspace is a generalization, this scheme shows how to define orthogonal projection onto any subspace of $$\mathbb{R}^n$$, of any dimension.

The two examples that we've seen since Definition 3.4 illustrate the first sentence in that definition. The next result justifies the second sentence.

We can find the orthogonal projection onto a subspace by following the steps of the proof, but the next result gives a convienent formula.

Exercises
/Solutions/