7.7 Gram–Schmidt Orthogonalization Process
INTRODUCTION
In Section 7.6 we saw that a vector space V can have many different bases. Recall, the defining characteristics of any basis B = {x1, x2, … , xn} of a vector space V is that
- the set B is linearly independent, and
- the set B spans the space.
In this context the word span means that every vector in the space can be expressed as a linear combination of the vectors x1, x2, … , xn. For example, every vector u in Rn can be written as a linear combination of the vectors in the standard basis B = {e1, e2, … , en}, where
e1 = 〈1, 0, 0, … , 0〉, e2 = 〈0, 1, 0, … , 0〉, …, en = 〈0, 0, 0, … , 1〉.
This standard basis B = {e1, e2, … , en} is also an example of an orthonormal basis; that is, the e
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