Proposition
Let
be a vector space with an orthonormal basis . Then for any vector the coefficient in front of in the decomposition of in will be . As a formula,
Proof
Let
. ==How to use orthonormality of
?== We need to use the fact that
and are orthogonal to each other (if and do not coincide). At the same time, we would like to isolate the summand corresponding to in the basis expansion. Therefore, let’s take the dot product of both sides with : (We used that
if , and .) Therefore, which is exactly what we were seeking to show.
More formulaically, the proposition allows us to explicitly write down expansion of vectors using the dot product given orthonormality of the basis.
Proposition
Let
be a vector space with an orthonormal basis . Then