Show that if T(x)=Ax preserve length and angles, then the two column vectors v and w must be...

A) Denote ~e1 and ~e2 as the standard vectors in R
2
. (See page 48 of the text for denitions.)
Then by Theorem 2.1.2 on page 47, we have T(~e1) = ~v and T(~e2) = ~w. Rather explicitly, denote
A =

a11 a12
a21 a22

=

~v ~w

in terms of ~v =

a11
a21

and ~w =

a12
a22

:
Using the denitions
~e1 =

1
0

and ~e2 =

0
1

we have
T(~e1) = A ~e1 = 1 ~v + 0 ~w = ~v;
T(~e2) = A ~e2 = 0 ~v + 1 ~w = ~w:
The standard vectors are perpendicular unit vectors. That is, both ~e1 and ~e2 are unit vectors, and
they are perpendicular to each other:
jj~e1jj
2
= 1
2
+ 0
2
= 1; jj~e2jj
2
= 0
2
+ 1
2
1; and ~e1 ~e2 = 1 0 + 0 1 = 0:
Since T preserves length, both ~v = T(~e1) and ~w = T(~e2) are also unit vectors. Since T preserves
angles, both ~v = T(~e1) and ~w = T(~e2) are also perpendicular to each other..