Orthonormal Bases

Abstract

Let V be a vector space. Let B₁, ..., B _n be a basis of V. To each vector X, we let correspond a set of scalars x₁, ..., x _n called the coordinates of X relative to the basis B₁, ..., B _n , by putting:

$$ X\;{\rm{ = }}\;{x_1}{B_1}\;{\rm{ + }}\;{x_2}{B_2}\;{\rm{ + }}\; \cdots \;{\rm{ + }}\;{x_n}{B_n}. $$