Given T IR1 right arrow IR2 defined by Tx y 3x 2y x y Find

Given T: IR^1 right arrow IR^2 defined by T(x, y) = (3x + 2y, x +y). Find formulas for To T(x, y) and T^-1(x, y).

Solution

1) ToT(x, y) = T(T(x, y)) = T(3x+2y, x+y) = T(a, b) (where a = 3x+2y and b=x+y).

= (3a+2b, a+b) = (3[3x+2y]+2[x+y], [3x+2y]+[x+y]) = (11x+8y, 4x+3y).

2) Let T^-1(x, y) = (a, b). Then (x, y) = T(a, b) = (3a+2b, a+b).

Therefore x = 3a+2b and y = a+b. Now, multiply the second equation by \'-2\' and then adding to first equation, we get a = x-2y. Now substitute \'a\' in y = a + b. Then we get y = x-2y+b. Implies that b = -x +3y.

Hence T^-1(x, y) = (a, b) = ( x-2y, -x +3y).

i. e., T^-1(x, y) = ( x-2y, -x +3y).