§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)