§18]
PRODUCTS OF LINEAR TRANSFORMATIONS
33
EXERCISES
1. The invariant aoa' 2 -\-a 2 a' 0 —2cha\ of
aox 2 +2aixy +a 2 y 2 , a'ox 2 +2a\xy +a' 2 y 2
is of total weight 2, but is not of constant weight in a 0 , ai, a 2 alone.
2. Verify the theorem for the Jacobian of two binary linear forms.
3. Verify the theorem for the Hessian of a ternary quadratic form.
4. No binary form of odd order p has an invariant of odd degree d.
18. Products of Linear Transformations. The product TV of
T :
V:
X=a£+Pv, y = y£+òri )
t = a'X+p'Y, 77 = 7 'X+S'Y,
5^0,
is defined to be the transformation whose equations are obtained
by eliminating £ and ?? between the equations of the given
transformations. Hence
I x=a"X+p"Y, y = y"X+8"Y,
[oT = aa -\~Py', P" = aft' P S', y" ~yY 8y f , 8" = yP'88'.
Its determinant is seen to equal AA' and hence is not zero.
By solving the equations which define T, we get
„ 8 P
P = -x—y,
A A
— 7 a
77= -x-\—y.
A A
These equations define the transformation T~ l inverse to T;
each of the products TT~ l and T~ 1 T is the identity trans
formation x = X, y=Y.
The product of transformation T g , defined in § 1, by 7V is seen to equal
T e+g in accord with the interpretation given there. The inverse of
T g is
T- g : £ = x cos 9+y sin 9, v= — x sin 9+y cos d.
Consider also any third linear transformation
T\i X =a.\U -\~P\V, Y—y\U-\-8iV.
To prove that the associative law
{TV)Ti = T{T'Ti)