Orthogonal Vectors and Othogonal Complements

Author:Ku, Yin Bon (Albert)

Orthogonal vectors

We generalize the notion of perpendicularity to

as follows: Definition: Vectors

and

are said to be orthogonal (to each other) if

, usually denoted by

. Observe that the zero vector is orthogonal to every vector in

. It can be shown that for any vectors

and

. If

, then

, which can be regarded as the Pythagorean theorem for

. By using the Pythagorean theorem repeatedly, we have the following result: Suppose

are

vectors in

such that they are orthogonal to each other i.e.

whenever

. Then

In the applet below, you will see why

is exactly the Pythagorean theorem in

. Also, the inner product and the angle

between the two vectors are related by the following formula:

Orthogonal Complements

Suppose

is a subspace of

. A vector

is said to be orthogonal to

for every

. Definition: The subset

is called the orthogonal complement of

Exercise

Prove that is a subspace of .

Suppose . Prove that is in if and only if for all .

In the applet below, suppose

. If

is a plane through the origin such that all vectors containing in the plane are orthogonal to

In general, give an m x n matrix

, we let

row vectors in

. Then consider the homogeneous equation

. Suppose

is in

i.e. a solution to this homogeneous equation. Then it is equivalent to saying that

for

i.e.

is orthogonal to

. Hence, we have

As the above result is true for all matrices, we apply it to

and get

It is obvious that

. Therefore, we have

Exercise

Find the orthogonal complement of in .

Orthogonal Vectors and Othogonal Complements