Suppose T: R^2->R^2 is a linear transformation with T(1,2)=(2,-1) & T(3,-1)=(-1,2). What is T(-12,11)?
To find the image of T(-12,11) under the linear transformation T, we can use the linearity property of linear transformations.
First, let's express the vector (-12,11) as a linear combination of the basis vectors (1,0) and (0,1) in R^2:
(-12,11) = -12*(1,0) + 11*(0,1)
Next, using the linearity property of T, we can compute T(-12,11) as follows:
T(-12,11) = T(-12*(1,0) + 11*(0,1))
Since T is a linear transformation, we can distribute T over the linear combination:
T(-12,11) = -12*T(1,0) + 11*T(0,1)
Now, we need to find the images of the basis vectors (1,0) and (0,1) under T. From the given information, we know that:
T(1,2) = (2,-1) and T(3,-1) = (-1,2)
Therefore, we can compute:
T(1,0) = T((1/3)*(3,0)) = (1/3)*T(3,0) = (1/3)*T((1,2)-(3,-1)) = (1/3)*(T(1,2) - T(3,-1)) = (1/3)*((2,-1)-(-1,2)) = (1/3)*(3,-3) = (1,-1)
T(0,1) = T((2/3)*(-3,1)) = (2/3)*T(-3,1) = (2/3)*T(-3*(1,2)+(3,-1)) = (2/3)*(-3*T(1,2) + T(3,-1)) = (2/3)*(-3*(2,-1)+(3,-1)) = (2/3)*(-9,-1) = (-6/3,-2/3) = (-2,-2/3)
Substituting these values into the equation for T(-12,11), we have:
T(-12,11) = -12*(1,-1) + 11*(-2,-2/3)
Now, we can compute the scalar multiplication and addition:
T(-12,11) = (-12,-12) + (-22,-22/3)
Adding the corresponding components, we get:
T(-12,11) = (-12 + (-22), -12 + (-22/3)) = (-34, -46/3)
Therefore, T(-12,11) = (-34, -46/3) is the image of (-12,11) under the given linear transformation T.
To find T(-12, 11), we can use linearity of the transformation:
First, let's find the expression of T in terms of a matrix. Since T is a linear transformation from R^2 to R^2, it can be represented by a 2x2 matrix.
Let's assume that T is represented by the matrix A. Then we can write:
T(x, y) = A * (x, y)
where (x, y) is a column vector.
Since T(1, 2) = (2, -1), we have:
A * (1, 2) = (2, -1)
Similarly, since T(3, -1) = (-1, 2), we have:
A * (3, -1) = (-1, 2)
We can write these equations in matrix form:
A * |1 3| = | 2 -1|
|2 -1| |-1 2|
To solve for matrix A, we can invert the left-hand side and multiply it with the right-hand side:
A = |1 3|⁻¹ * | 2 -1|
|2 -1| |-1 2|
To find the inverse of a 2x2 matrix, we can use the formula:
|a b|⁻¹ = 1/(ad - bc) * | d -b|
|-c a|
Therefore:
A = 1/(1*(-1) - 3*2) * | -1 -3| * | 2 -1|
| -2 1| |-1 2|
Simplifying this expression gives:
A = 1/(-7) * | -1 -3| * | 2 -1|
| -2 1| |-1 2|
A = 1/(-7) * | -7 4|
| -4 1|
A = | 1 -4/7|
| 4/7 -1|
Now we can find T(-12, 11) by multiplying A with (-12, 11):
T(-12, 11) = | 1 -4/7| * (-12, 11)
| 4/7 -1|
Let's calculate this multiplication:
T(-12,11) = (1*(-12) + (-4/7)*11, (4/7)*(-12) + (-1)*11)
= (-12 - 44/7, -48/7 - 11)
= (-84/7 - 44/7, -48/7 - 77/7)
= (-128/7, -125/7)
Therefore, T(-12, 11) is equal to (-128/7, -125/7).