Let T: R2 --> R2 be the linear transformation given by T(x,y)=(2x+y, x+y). Find T -1(x,y).Select one:a. (x-y, -x+2y)b. (x-y, x+2y)c. None of themd. (x+y, -x+2y)
Question
Let T: R2 --> R2 be the linear transformation given by T(x,y)=(2x+y, x+y). Find T -1(x,y).Select one:a. (x-y, -x+2y)b. (x-y, x+2y)c. None of themd. (x+y, -x+2y)
Solution
To find the inverse of the transformation T, we need to solve the system of equations:
2x + y = u x + y = v
where (u, v) is the image under T of (x, y). We want to find (x, y) in terms of (u, v).
Subtract the second equation from the first to get x = u - v.
Substitute x = u - v into the second equation to get u - v + y = v, which simplifies to y = v - u + v = 2v - u.
So, the inverse transformation T^-1(u, v) = (u - v, 2v - u).
Therefore, the correct answer is a. (x-y, -x+2y).
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