Question

Let T: R³ → R² be a linear transformation. Find T(8, 3, 7) if T(1, 0, -1) = (2, 3) and T(2, 1, 3) = (-1, 0).

Answer 1

To find T(8, 3, 7), we can express the vector (8, 3, 7) as a linear combination of the given vectors (1, 0, -1) and (2, 1, 3). Solving the resulting equations, we obtain T(8, 3, 7) = (7, 5, 2).

Step-by-step explanation:

To find T(8, 3, 7), we can use the linearity property of the linear transformation, along with the given information. First, we can write the vector (8, 3, 7) as a linear combination of the vectors (1, 0, -1) and (2, 1, 3).

1. Let's find the coefficients a and b in the equation:

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

2. Using the given information:

   
   a(1, 0, -1) + b(2, 1, 3) = (8, 3, 7)

3. From the first equation:

   
   a + 2b = 8

4. From the second equation:

   
   -b + 3b = 3

5. Solving these equations:

   
   a = 5, b = 1

6. Substituting these values back in the equation:

Therefore, T(8, 3, 7) = (7, 5, 2).