The transformation T:R^2--》R^2 is defined by T(1,0)=(2,-5) and T(-3,2)=(1,4). Find the natural matrix of the linear transformation T and find T(x,y).

To find the natural matrix of the linear transformation T, we need to determine the images of the standard basis vectors (1, 0) and (0, 1) under T.

Given that T(1, 0) = (2, -5) and T(-3, 2) = (1, 4), we can write the following equations:

T(1, 0) = a(1, 0) + b(0, 1)
T(-3, 2) = c(1, 0) + d(0, 1)

We can rewrite these equations as a system of linear equations:

2 = a
-5 = b
1 = c
4 = d

Therefore, the coefficients of the matrix representing T are:
[[2, -5],
[1, 4]]

To find T(x, y), we can write the following equation:

T(x, y) = a(x, y) + b(0, 1)

Using the coefficients we found earlier, we have:

T(x, y) = 2(x, y) + (-5)(0, 1)

Simplifying, we get:

T(x, y) = (2x, 2y) + (0, -5)

Therefore, T(x, y) = (2x, 2y - 5).