Final answer:
To find T(-1-4x), we must first solve for T(1) and T(x) using the given linear transformation equations, then use these results to compute T(-1-4x).
Step-by-step explanation:
To determine T(-1-4x), we observe that the given information forms a system of linear equations based on the linear transformation T. The transformation of 1 and x must be determined to express any linear combination of these basis elements. We have two equations, T(1+5x)=1+2x and T(5+24x)= -2-3x, which can be decomposed into T(1) + 5T(x) and 5T(1) + 24T(x), respectively. Solving for T(1) and T(x) from these equations will provide us with enough information to find T(-1-4x).
Let A = T(1) and B = T(x). The given transformations become A + 5B = 1 + 2x and 5A + 24B = -2 - 3x. Solving these equations simultaneously will give us the coefficients A and B that define T. Once found, we can use these coefficients to find the transformed vector of any linear combination of 1 and x, particularly T(-1-4x).
Since the details from the provided reference do not directly apply to solving the given linear transformation problem, they are not used in computing the answer.