Orthogonal Projection
Orthogonal Sets
A set of vectors in is said to be an orthogonal set if whenever .
Orthogonal sets enjoy some really nice properties. The following is one of them:
Theorem: Any orthogonal set of nonzero vectors is a linearly independent set.
Proof: Let be an orthogonal set of nonzero vectors in . Suppose there exist real numbers such that
Take the inner product with , where on both sides, we get
By definition, whenever . Therefore, we have
Since , for , which implies that the orthogonal set is a linearly independent set.
Definition: An orthogonal basis for a subspace of is a basis for that is also an orthogonal set.
An orthogonal basis is more convenient to use than other bases because when a vector is expressed as a linear combination of an orthogonal basis, the weights can be computed easily using inner products.
Theorem: Let be an orthogonal basis for a subspace of . For any in such that
,
the weights can be computed by the formula: , for .
Prove the above formula for . (Hint: Use the same idea as in the proof of the previous theorem).
Orthogonal Projection
We take a closer look at the formula for the weights: , for . Let be the unit vector in the direction of and be the line containing . In other words, . Recall the projection map , which is the signed distance from the origin to the point obtained by projecting the arrowhead of to orthogonally. Hence, is exactly the vector from the origin to that projection point. This vector is said to be the orthogonal projection of onto , denoted by . Then we have
In other words, is regarded as the sum of all orthogonal projection of onto for .
In the applet below, you can change the vector and the orthogonal basis and see the orthogonal projection of onto the line spanned by for .