okay
in this question we have a linear map
which we'll simply reflect vector in the
line y equals x which is the dotted line
here and we want to show that the vector
1 1 is an eigenvector find the eigen
value and the same thing for the vector
1 -1 so let's figure out where
those vectors are if this is 1 here and
1 up here then the vector 1 1 lies on this
axis of reflection while the vector 1
-1 is 1 across -1 down
pointing down here 1 -1 is that vector now what's the
reflection going to do to the vector 1 -1
nothing it's already on the axis of
reflection so we see that if you apply a
T to 1 1 you just get the vector 1 1 back
it's 1 1 back again and we want to write
this as a constant times 1 1 obviously
the constant is 1 so the vector 1 1 is
an eigenvector with eigenvalue 1
is an eigenvector of T to be specific with eigenvalue 1
and similarly now what happens if we reflect
the vector 1 -1 in this line y
equals x you can see instead of being
down in this quadrant it's gonna flip
over here and we'll get the vector which
is -1 in the x coordinate and 1 in
the y coordinate okay
so this one is T of 1-1, T of  1-1 as
we've seen from the geometric picture
it's -1 1 and again we want to
write this as a scalar multiple of the
vector we started with obviously that
scalar is -1 and this says that the
vector 1 -1 is an eigen vector of T
with eigenvalue -1
and that completes the question or part
of the question
while we're here we may
as well find the matrix that represents
this map and we're going to do that
using the matrix representation theorem
the map T goes from R2 to R2 so the
matrix a will be A 2 by 2 matrix the
first column equals T of the first
standard basis vector 1 0 so here is the
first standard basis vector 1 0 what
happens to this vector under T well when
you reflect it'll reflect up to sorry
for the mess the second standard basis
vector 0 1 T of 1 0 is 0 1
so we note that T of 1 0 equals 0 1
and now if you perform a reflection
twice you'll end up back where you were
so we can see that T of 0 1 will take us
back to 1 0
that was easy, we only really had to calculate one thing and the matrix
representation theorem says that these
vectors are our two columns of A so
without any further calculation we just
write down A the first column is T of 1
0 which in this example is 0 1 the
second column is T of 0 1 which here is
1 0 and that's it
