Orthonomal Basis

n theorem 8.1.5 we saw that every set of nonzero orthogonal vectors is linearly independent. This motivates our next definition.

8.2.1 Definition Let V be an inner product space and W a subspace of V. An orthonormal set is called an orthonormal basis of W if . As an immediate application of theorem 8.1.5, we have the following :

8.2.3 Example: For V = with usual inner-product, the set with ,where 1 is at the place, is an orthonormal basis of V. This is called standard basis of .

8.2.4 Gram-Schmidt algorithm: Based on the proof of theorem 8.1.5(iv), the procedure for finding an orthonormal basis for an inner product space is given as follows: Let be a basis of V. Step1: Define Step 2: : Step 3: Define Then, is an orthonormal basis of V .

8.2.5 Examples: (i) In example 8.1.6, the basis S={(1, 1, 1, ), (-1, 0, -1), (-1, 2, 3)} of gave the orthonormal basis (ii) Consider the vector space V = of all real-polynomial function p(x), ,of degree at most 3. Let the inner-product on V be given by Consider the basis S = for V. The Gram-Schmidt process gives the following. Next, define Then is an orthonormal basis of V. (iii) Let W denote the subspace of spanned by the set of vectors: S = {(1, 1, 1), (2, 2, 3), (0, 0, 1), (1, 2, 3)}. Clearly S is not a basis of W , since S has four elements. Thus, to find an orthonormal basis of W one way is to first select a basis of W out of the vectors in S and then apply Gram-Schmidt process to it. Another, more straight forward method, is to apply Gram-Schmidt process directly to the set of vectors in S , and discard those vectors which become zero. In our case it is as follows: The required orthonormal basis is

8.2.6 Note: Suppose we want to find an orthonormal basis of which includes a given unit vector Of course, one method is first extend {X} to a basis of and then using Gram-Schmidt process orthonormalize it. An efficient way of doing this is as follows: (i) Write Assume (ii) Construct (iii) It is easy to check that the matrix U has the property i.e. Columns of U are orthonormal.