What is the Gram-Schmidt process used for?

In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product.

What is meant by Gram-Schmidt orthogonalization process?

Gram-Schmidt orthogonalization, also called the Gram-Schmidt process, is a procedure which takes a nonorthogonal set of linearly independent functions and constructs an orthogonal basis over an arbitrary interval with respect to an arbitrary weighting function .

What are the conditions that allow us to use the output vectors from the Gram-Schmidt algorithm to form a basis?

The vectors to which the Gram-Schmidt process is applied must be of the same dimension and orientation, and the orthonormalized vectors will be in the same orientation as the input set.

Why Gram Schmidt orthogonalization process is required?

We now come to a fundamentally important algorithm, which is called the Gram-Schmidt orthogonalization procedure. This algorithm makes it possible to construct, for each list of linearly independent vectors (resp. basis), a corresponding orthonormal list (resp. orthonormal basis).

What is Gram Schmidt orthogonalization procedure in digital communication?

In Digital communication, we apply input as binary bits which are converted into symbols and waveforms by a digital modulator. These waveforms should be unique and different from each other so we can easily identify what symbol/bit is transmitted. To make them unique, we apply Gram-Schmidt Orthogonalization procedure.

Why is modified Gram-Schmidt better?

Modified Gram-Schmidt performs the very same computational steps as classical Gram-Schmidt. In classical Gram-Schmidt you compute in each iteration a sum where all previously computed vectors are involved. In the modified version you can correct errors in each step.

How do you prove an orthonormal basis?

Definition: A basis B = {x1,x2,…,xn} of Rn is said to be an orthogonal basis if the elements of B are pairwise orthogonal, that is xi · xj whenever i = j. If in addition xi · xi = 1 for all i, then the basis is said to be an orthonormal basis.

Why is modified Gram Schmidt better?