To find the value of T(7,8), we can use the given information about the linear transformation T and apply it to the vector (7,8).



A linear transformation T: R² → R² is a function that takes a vector in R² (a 2-dimensional space) as input and maps it to another vector in R². In other words, it transforms one vector to another vector in the same 2-dimensional space.

The transformation T can be represented by a 2x2 matrix called the transformation matrix. Let's denote the transformation matrix as A:

A = [a b]
[c d]

The matrix A represents the coefficients of the linear transformation. Each column of A corresponds to the image of the standard basis vectors i and j.



We are given two points and their corresponding images under the linear transformation T:

T(3,1) = (2,-4)
T(1,1) = (0,2)

Using these points, we can set up a system of equations to find the values of the transformation matrix A:

a(3) + b(1) = 2
c(3) + d(1) = -4
a(1) + b(1) = 0
c(1) + d(1) = 2

Simplifying these equations, we get:

3a + b = 2 --> Equation 1
3c + d = -4 --> Equation 2
a + b = 0 --> Equation 3
c + d = 2 --> Equation 4

Solving this system of equations, we find:

a = -1
b = 1
c = 3
d = -7

Therefore, the transformation matrix A is:

A = [-1 1]
[3 -7]



Now that we have the transformation matrix A, we can find the image of the vector (7,8) under the linear transformation T.

To do this, we multiply the transformation matrix A by the vector (7,8):

A * [7] = [-1 1] * [7] = [-1(7) + 1(8)] = [-7 + 8] = [1]
[8] [3 -7] [8] [3(7) - 7(8)] [21 - 56] [-35]

Therefore, T(7,8) = (1,-35).

Final Answer: T(7,8) = (1,-35)