Orthogonal sets - Orthogonality and Least Squares
Orthogonal sets
Lessons
Notes:
A set of vectors {} in are orthogonal sets if each pair of vectors from the set are orthogonal. In other words,
Where .
If the set of vectors {} in is an orthogonal set, then the vectors are linearly independent. Thus, the vectors form a basis for a subspace . We call this the orthogonal basis.
To check if a set is an orthogonal basis in , simply verify if it is an orthogonal set.
Are calculated by using the formula:
where .
A set {}is an orthonormal set if it's an orthogonal set of unit vectors.
If is a subspace spanned by this set, then we say that {} is an orthonormal basis. This is because each of the vectors are already linear independent.
A matrix has orthonormal columns if and only if .
Let be an matrix with orthonormal columns, and let and be in . Then the 3 following things are true:
1)
2)
3) if and only if
Consider to be the subspace spanned by the vector . Then the orthogonal projection of onto is calculated to be:
proj
The component of orthogonal to (denoted as ) would be:
-
Intro Lesson
Orthogonal Sets Overview:
