Linear Algebra, 2020a

322 Chapter Three. Maps Between Spaces
often use this term to mean not just that the columns are orthogonal but also
that they have length one).
We can leverage this characterization to understand the geometric actions
of distance-preserving maps. Because ‖t(⃗v )‖ = ‖⃗v ‖, the map t sends any ⃗v
somewhere on the circle about the origin that has radius equal to the length of
⃗v. In particular, ⃗e 1 and ⃗e 2 map to the unit circle. What’s more, once we fix the
unit vector ⃗e 1 as mapped to the vector with components a and b then there
are only two places where ⃗e 2 can go if its image is to be perpendicular to the
first vector’s image: it can map either to one where ⃗e 2 maintains its position a
quarter circle clockwise from ⃗e 1
( ) −b
a
( a
b)
(
a
Rep E2 ,E 2
(t) =
b
)
−b
a
or to one where it goes a quarter circle counterclockwise.
( a
b)
( ) b
−a
(
a
Rep E2 ,E 2
(t) =
b
)
b
−a
The geometric description of these two cases is easy. Let θ be the counterclockwise
angle between the x-axis and the image of ⃗e 1 . The first matrix above
represents, with respect to the standard bases, a rotation of the plane by θ
radians.
( ) −b
a
( a
b)
( )
x t
↦−→
y
(
)
x cos θ − y sin θ
x sin θ + y cos θ
The second matrix above represents a reflection of the plane through the line
bisecting the angle between ⃗e 1 and t(⃗e 1 ).