Linear Algebra

288 Chapter 3. Maps Between Spaces
lengths of vectors, because then for all �p and �q the distance between the two is
preserved �t(�p − �q )� = �t(�p) − t(�q )� = ��p − �q �. For that check, let
�v =
� �
x
y
t(�e1) =
� �
a
b
t(�e2) =
� �
c
d
and, with the ‘if’ assumptions that a 2 + b 2 = c 2 + d 2 =1andac + bd =0we
have this.
�t(�v )� 2 =(ax + cy) 2 +(bx + dy) 2
= a 2 x 2 +2acxy + c 2 y 2 + b 2 x 2 +2bdxy + d 2 y 2
= x 2 (a 2 + b 2 )+y 2 (c 2 + d 2 )+2xy(ac + bd)
= x 2 + y 2
= ��v � 2
One thing that is neat about this characterization is that we can easily
recognize matrices that represent such a map with respect to the standard bases.
Those matrices have that when the columns are written as vectors then they
are of length one and are mutually orthogonal. Such a matrix is called an
orthonormal matrix or orthogonal matrix (the second term is commonly used to
mean not just that the columns are orthogonal, but also that they have length
one).
We can use this insight to delimit the geometric actions possible in distancepreserving
maps. Because �t(�v )� = ��v �, any�v is mapped by t to lie somewhere
on the circle about the origin that has radius equal to the length of �v, and
so in particular �e1 and �e2 are mapped to vectors on the unit circle. What’s
more, because of the orthogonality restriction, once we fix the unit vector �e1 as
mapped to the vector with components a and b then there are only two places
where �e2 can go.
� � −b
a
� � a
b
Thus, only two types of maps are possible.
Rep E2,E2 (t) =
� �
a −b
b a
� � a
b
� � b
−a
Rep E2,E2 (t) =
� �
a b
b −a
We can geometrically describe these two cases. Let θ be the angle between the
x-axis and the image of �e1, measured counterclockwise.
The first matrix above represents, with respect to the standard bases, a
rotation of the plane by θ radians.