Basis and dimension
Definition 1. We say that vectors form a basis in a subspace if 1) it is spanned by and 2) these vectors are linearly independent. The number of vectors in the basis is called a dimension of and the notation is
An orthogonal basis is a special type of a basis, when in addition to the above conditions 1)-2) the basis vectors are orthonormal. For the dimension definition to be correct, the number of vectors in any basis should be the same. We prove correctness in a separate post.
Direct proof. Any can be represented as
If the right side is zero, then and all are zero.
Proof using orthonormality. If the right side in (1) is zero, then for all
Proof. (1) shows that is spanned by Besides, they are linearly independent by Exercise 1.
Definition 2. Let be two subspaces such that any element of one is orthogonal to any element of the other. Then the set is called an orthogonal sum of and denoted
Exercise 3. If a vector belongs to both terms in the orthogonal sum of two subspaces , then it is zero. This means that
Proof. This is because any element of is orthogonal to any element of so is orthogonal to itself, and
Exercise 4 (dimension additivity) Let be an orthogonal sum of two subspaces. Then
Proof. Let By definition, is spanned by some linearly independent vectors and is spanned by some linearly independent vectors Any can be decomposed as Since can be further decomposed as the system spans
Moreover, this system is linearly independent. If
By Exercise 3 then By linear independence of the vectors in the two systems all coefficients must be zero.
The conclusion is that