Page 262 - 35Linear Algebra

262 Orthonormal Bases and Complements

u
v
u·v u = v k
u·u

v ⊥

⊥
This new vector v is orthogonal to u because
u · v
⊥
u v = u v − u u = 0.
u · u
⊥
Hence, {u, v } is an orthogonal basis for span{u, v}. When v is not par-
n ⊥ o
allel to u, v ⊥ 6= 0, and normalizing these vectors we obtain u , v , an
⊥
|u| |v |
orthonormal basis for the vector space span {u, v}.
⊥
k
Sometimes we write v = v + v where:
u · v
v ⊥ = v − u
u · u
u · v
v k = u.
u · u
This is called an orthogonal decomposition because we have decomposed
v into a sum of orthogonal vectors. This decomposition depends on u; if we
k
⊥
change the direction of u we change v and v .
⊥
⊥
3
If u, v are linearly independent vectors in R , then the set {u, v , u×v }
3
would be an orthogonal basis for R . This set could then be normalized by
dividing each vector by its length to obtain an orthonormal basis.
However, it often occurs that we are interested in vector spaces with di-
mension greater than 3, and must resort to craftier means than cross products
2
to obtain an orthogonal basis .
2
Actually, given a set T of (n − 1) independent vectors in n-space, one can define an
analogue of the cross product that will produce a vector orthogonal to the span of T, using
a method exactly analogous to the usual computation for calculating the cross product
3
of two vectors in R . This only gets us the “last” orthogonal vector, though; the Gram–
Schmidt process described in this section gives a way to get a full orthogonal basis.

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