The Gram-Schmidt Process Conceptualized
The best way understand the inner workings of the Gram-Schmidt orthonormalization process is to use orthogonal splittings of vectors as follows.
Theorem (Gram-Schmidt Orthonormalization Process) Given a linearly independent subset 
{a
, ... ,a
} of 
, the set of vectors 
{v, ... ,v} defined below is an ONB of 
span().
{a, ... ,a} of , the set of vectors {v, ... ,v} defined below is an ONB of span().Let 
span{a, ... ,a}, and define recursively by
Thus each v
results from normalizing the component of a perpendicular to . As a result, {v, ... ,v} is an ONB for , for each 1 
.
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