If T:ℝ2→ℝ2 is a transformation and the action of T is as given, find a formula for T−1(V), where V is any vector in ℝ2.
T[x] = [x + 3y]
[y] = [-2x - 5y]
T^-1 = [x] = ?
[y]
To find the formula for T^(-1)(V), we need to find the inverse transformation of T.
To do that, we can set up a system of equations by equating the components of T[x] with the components of [x] and [y].
From the given action of T:
x + 3y = x
-2x - 5y = y
Simplifying the first equation, we get:
3y = 0
y = 0
Substituting y = 0 into the second equation:
-2x = y
-2x = 0
x = 0
Therefore, the solution to the system of equations is x = 0 and y = 0.
Thus, the formula for T^(-1)(V) is:
T^(-1)(V) = [0]
[0]