Orthonormal bases

01An orthonormal basis is a set of perpendicular directions, each of length 1#

An orthonormal basis is a set of perpendicular directions, each of length 1. These two properties let dot products report coordinates directly. They are what make the coordinate changes in QR, SVD, and PCA especially convenient.

Check both properties separately: dot each vector with itself to check length, and dot different vectors to check perpendicularity. Perpendicular vectors of length 2 are orthogonal, but they are not orthonormal.

An orthonormal basis uses perpendicular unit vectors. For the supplied basis $u=(0.6,0.8)$ and $v=(-0.8,0.6)$, check both lengths and their dot product.

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For $u=(0.6,0.8)$ and $v=(-0.8,0.6)$, each squared length is $0.36+0.64=1$. Their dot is $-0.48+0.48=0$. These values meet both orthonormal conditions.

For $u=(0.6,0.8)$ and $v=(-0.8,0.6)$, squared lengths are $0.36+0.64=1$ for each. Their dot product is $-0.48+0.48=0$. These directions can therefore be used as unit perpendicular axes. A coordinate along $u$ is obtained by $x\cdot u$.

02Unit length alone is also insufficient#

Unit length alone is also insufficient. The basis $(1,0)$, $(0.6,0.8)$ has two unit vectors, but their dot is 0.6. With three vectors, check all three distinct pairs.

03A supplied basis already spans its space#

A supplied basis already spans its space. To verify orthonormality, check each length and each distinct pair. Negative coordinates and rotated directions are allowed. A zero vector cannot have unit length.

Further questions