A set of vectors {

$v_1,\cdots,v_n$} in

$\Bbb{R}^n$ are

**orthogonal sets** if each pair of vectors from the set are orthogonal. In other words,

$v_i \cdot v_j =0$
Where

$i \neq j$.

If the set of vectors {

$v_1,\cdots,v_n$} in

$\Bbb{R}^n$ is an orthogonal set, then the vectors are linearly independent. Thus, the vectors form a basis for a subspace

$S$. We call this the

**orthogonal basis**.

To check if a set is an orthogonal basis in

$\Bbb{R}^n$, simply verify if it is an orthogonal set.

$y=c_1 v_1+c_2 v_2+\cdots+c_p v_p$
Are calculated by using the formula:

$c_i = \frac{y \cdot v_i}{v_i \cdot v_i}$
where

$i=1,\cdots,p$.

A set {

$v_1,\cdots,v_p$}is an

**orthonormal set** if it's an orthogonal set of unit vectors.

If

$S$ is a subspace spanned by this set, then we say that {

$v_1,\cdots,v_p$} is an

**orthonormal basis**. This is because each of the vectors are already linear independent.

A

$m \times n$ matrix

$U$ has orthonormal columns if and only if

$U^T U=I$.

Let

$U$ be an

$m \times n$ matrix with orthonormal columns, and let

$x$ and

$y$ be in

$\Bbb{R}^n$. Then the 3 following things are true:

1)

$\lVert Ux \rVert = \lVert x \rVert$
2)

$(Ux) \cdot (Uy)=x \cdot y$
3)

$(Ux) \cdot (Uy)=0$ if and only if

$x \cdot y =0$
Consider

$L$ to be the subspace spanned by the vector

$v$ . Then the orthogonal projection of

$y$ onto

$v$ is calculated to be:

$\hat{y}=$proj$_Ly=\frac{y \cdot v}{v \cdot v}v$
The component of

$y$ orthogonal to

$v$ (denoted as

$z$) would be:

$z=y-\hat{y}$