Orthogonal Sets and Bases

Orthogonal Set

Definition

A set is an orthogonal set if for .

Intuition

An orthogonal set is simply a set of vectors which are all orthogonal (90 degrees) to eachother.

Example

The standard basis for is an orthogonal set. Also, the set

is an orthogonal set in .

Remark

An orthogonal set may contain the zero vector, and any set containing the zero vector is linearly dependent. Otherwise, it is linearly independent.

Theorem

If is an orthogonal set of nonzero vectors, then is linearly independent.

Proof

For , consider . For each ,

Expanding the dot product on the left and evaluating the one on the right gives

Since is an orthogonal set, we have for . Thus we obtain

since and we must have . Since war arbitrary, it follows that which means is linearly independent.

Definition

If an orthogonal set is a basis for a subspace or , then is an orthogonal basis for .

Remark

We can calculate the coefficients used to create any arbitrary vector by projecting is onto each basis vector.

Intuition

If we have , an orthogonal basis of a subspace of . Then we have

That means each coefficient is

Visualization

We have the vector , and we have "found" the coefficients with some projections.

Example

Let and
be an orthogonal basis for . Write as a linear combination of the vectors in .

For , consider

then

and so