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G.13 Orthonormal Bases and Complements 421
G.13 Orthonormal Bases and Complements
All Orthonormal Bases for R 2
We wish to find all orthonormal bases for the space R 2 , and they are {e θ 1, e θ 2}
up to reordering where
e θ 1 =
( ) cos θ
, e
sin θ
θ 2 =
( ) − sin θ
,
cos θ
for some θ ∈ [0, 2π). Now first we need to show that for a fixed θ that the pair
is orthogonal:
Also we have
e θ 1 e θ 2 = − sin θ cos θ + cos θ sin θ = 0.
‖e θ 1‖ 2 = ‖e θ 2‖ 2 = sin 2 θ + cos 2 θ = 1,
and hence {e θ 1, e θ 2} is an orthonormal basis. To show that every orthonormal
basis of R 2 is {e θ 1, e θ 2} for some θ, consider an orthonormal basis {b 1 , b 2 } and
note that b 1 forms an angle φ with the vector e 1 (which is e 0 1). Thus b 1 = e φ 1 and
if b 2 = e φ 2 , we are done, otherwise b 2 = −e φ 2 and it is the reflected version.
However we can do the same thing except starting with b 2 and get b 2 = e ψ 1 and
b 1 = e ψ 2 since we have just interchanged two basis vectors which corresponds to
a reflection which picks up a minus sign as in the determinant.
-sin θ
cos θ
cos θ
sin θ
θ
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