GEOMETRIC TRANSFORMATIONS
           So far we've used vectors and matrices for the most part to write a system of linear equations in more concise
       form as a vector or matrix equation. This often enabled us to interpret such properties of systems of equations as
       unique or multiple solutions in different ways and to find these solutions systematically using row reduction, for
       instance. Systems of linear equations still occur but more as a step on the way to understanding or interpreting
       something rather than as something of importance in its own right.
          Previously we associated an m × n matrix A with a linear transformation
                                                               T (x)  =  Ax,                     T :Rn → Rm
                                                                 A                                 A
       using matrix-vector multiplication. And conversely, by Fundamental Theorem 1, each linear transformation 
       T : Rn  →  Rm can be written as T = T  where A is the Standard Matrix. But frequently, a linear transformation T
                                                                 A
       is described in geometric terms or by some mathematical property, say, as rotation through of prescribed angle.
       Let's see how this works for a number of geometric transformations T : R2 → R2. By the Fundamental Theorem all
       that we need do is determine T(e ) and T(e ) where e  and e  correspond to the usual i = (1, 0) and j = (0, 1)
                                                        1               2              1          2
       in the plane.
               Rotation through ϕ counter-clockwise about the origin:
              Since
                          T(e )  =  T([1])  =  [cosϕ],
                                1                0               sin ϕ
            and
                        T(e )  =  T([0])  =  [−sinϕ],
                              2                1                 cosϕ
            we get
                                 A  =  [cosϕ −sinϕ].
                                             sin ϕ       cosϕ
               Rotation through ϕ clockwise about the origin:
                       Since
                             T(e )  =  T([1])  =  [ cosϕ ],
                                   1                0               −sinϕ
                     and
                               T(e )  =  T([0])  =  [sinϕ],
                                     2                1               cosϕ
                     we get
                                      A  =  [ cosθ             sin θ ].
                                                  −sinθ cosθ
               Reflection through the x1-axis:
                       Since
                                 T(e )  =  T([1])  =  [1],
                                       1                0               0
                     and
                                T(e )  =  T([0])  =  [ 0 ],
                                      2                1               −1
                     we get
                                           A  =  [1          0 ].
                                                       0 −1
           Get the idea? Now try to establish these yourself: 
                 Reflection through the x2-axis:                                                      Reflection through the line x1 = x2:
                                      A  =  [−1 0].                                                                         A  =  [0         1].
                                                   0     1                                                                             1 0
                 Reflection through the line x1 + x2 = 0:                                             Reflection through the origin:
                                    A  =  [ 0           −1].                                                             A  =  [−1            0 ].
                                                −1       0                                                                            0      −1
           Since T       ∘T =T  for linear transformations, the standard matrix associated with compositions of geometric
                      A       B        AB
       transformations is just the matrix product AB.
              Problem : find the Standard matrix for the linear
            transformation T : R2 → R2 which first
                    rotates points counter-clockwise about the origin
            through π/4,
            and then
                    reflects points through the line x1 = x2.
              Solution: the action of T is shown graphically to the
            right. Now T is the composition T ∘ T  of the matrix
                                                            A       B
            transformation T  rotating R2 counter-clockwise
                                    B
            through π/4 about the origin and the matrix
            transformation T  reflecting R2 in the line x1 = x2
                                    A
            shown in purple, where                                                               Thus the Standard matrix for T = T ∘ T  is
                                                                                                                                                     A       B
                 A  =  [0         1],      B  =  [cosπ/4 −sinπ/4].                                   [0 1]( 1 [1 −1]) =  1 [1                                      1 ].
                             1 0                        sin π/4         cosπ/4                         1 0          √2 1 1                          √2 1 −1
           One perhaps surprising consequence of this matrix/geometric approach to linear transformations is that familiar
       trig identities can often be made completely natural and transparent. For suppose T  defines rotation counter-
                                                                                                                                θ
       clockwise about the origin through θ, and T  defines rotation counter-clockwise about the origin through ϕ. Then
                                                                      ϕ
       geometrically the composition of these two transformations is surely rotation counter-clockwise about the origin
       through θ+ϕ, i.e. T ∘ T                 = T         . But algebraically,
                                     θ      ϕ         θ+ϕ
             T ∘T   =  [cosθ −sinθ][cosϕ −sinϕ] = ⎡cosθcosϕ−sinθsinϕ −(cosθsinϕ+sinθcosϕ)⎤,
               θ     ϕ           sin θ      cosθ          sin ϕ       cosϕ              ⎣                                                                               ⎦
                                                                                           sin θcosϕ+cosθsinϕ                      cosθcosϕ−sinθsinϕ
       and so
                   T        =  [cos(θ + ϕ)            −sin(θ+ϕ)] = ⎡cosθcosϕ−sinθsinϕ −(cosθsinϕ+sinθcosϕ)⎤.
                     θ+ϕ           sin(θ + ϕ)           cos(θ+ϕ)                  ⎣                                                                              ⎦
                                                                                     sin θcosϕ+cosθsinϕ                      cosθcosϕ−sinθsinϕ
       Consequently,
                                cos(θ+ϕ) = cosθcosϕ−sinθsinϕ,                                   sin(θ + ϕ) = sinθcosϕ+cosθsinϕ,
       as you've probably known for a long time, but never perhaps known why!!
           Your Turn Now: for a geometric matrix transformation x ⟶ Ax of 3-space the 2 × 2 matrix A is replaced by
       a 3 ×3 matrix.
                    Is the projection of R3 onto the xy-plane a linear transformation? If so, what is its Standard matrix?
                    Determine A when T  is rotation through θ around the x-axis. (Distinguish between clockwise and counter-
                                                  A
       clockwise rotation.)
                    Determine B when T  is rotation through ϕ around the y-axis. What is the composition T ∘ T ?
                                                  B                                                                                                  A       B