A set of vectors in is an orthonormal set if it is an orthogonal set of unit vectors. It is an orthonormal basis for a subspace of if it spans . Let be an m x n matrix. Suppose its column vectors in form an orthonormal set. It is equivalent to because the -entry of is for and is an orthonormal set if and only if for and for . Here are some nice properties of : For any vector in , 1. 2. (Note: If we view be a linear transformation from to , then this linear transformation preserves the inner product and norm.) Proof: 1. By definition, and . Hence, 2. Using (1), In particular, when , is an n x n square matrix such that , which means is invertible and . Such a square matrix is called an orthogonal matrix. From above, we know that the linear transformation preserves the inner product and norm. Moreover, it also preserve volume because . (Why?) If is an orthonormal basis for a subspace of , then for any in , then it is very convenient to write down its orthogonal projection onto : If , then . Proof: Easy

Given a basis for a subspace of , we have a simple way called the Gram-Schmidt process to produce an orthogonal or orthonormal basis for the subspace. The idea is as follows: Suppose you are given a basis of a subspace of . Start with and let it be the first one in the orthogonal basis that we are going to produce and call it i.e. . Let . We consider the orthogonal decomposition and get the component in i.e. Then is an orthogonal set. Let . We consider the orthogonal decomposition and get the component in i.e. Then is an orthogonal set. We repeat this procedure. At the i^th step, we let and After p steps, we obtain the orthogonal basis for . If you want to make it an orthonormal basis, then you can simply normalize each vector to , where for . Remark: If you change the order of the vectors in the given basis, you will get a different orthogonal basis from the Gram-Schimdt process. Example: Find an orthonormal basis for in . You can try this online Gram-Schimdt calculator.