Page 421 - 35Linear Algebra

G.13 Orthonormal Bases and Complements 421

G.13 Orthonormal Bases and Complements

2
All Orthonormal Bases for R
θ
θ
2
We wish to find all orthonormal bases for the space R , and they are {e , e }
1 2
up to reordering where

cos θ − sin θ
θ
θ
e = , e = ,
1 sin θ 2 cos θ
for some θ ∈ [0, 2π). Now first we need to show that for a fixed θ that the pair
is orthogonal:
θ
e θ e = − sin θ cos θ + cos θ sin θ = 0.
1 2
Also we have

2
θ 2
θ 2
2
ke k = ke k = sin θ + cos θ = 1,
1 2
θ
θ
and hence {e , e } is an orthonormal basis. To show that every orthonormal
1 2
2
θ
θ
basis of R is {e , e } for some θ, consider an orthonormal basis {b 1 , b 2 } and
1 2
φ
0
note that b 1 forms an angle φ with the vector e 1 (which is e ). Thus b 1 = e and
1 1
φ φ
if b 2 = e , we are done, otherwise b 2 = −e 2 and it is the reflected version.
2
ψ
However we can do the same thing except starting with b 2 and get b 2 = e 1 and
ψ
b 1 = e since we have just interchanged two basis vectors which corresponds to
2
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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