Question

Let r: R² → R² be a linear transformation that maps (5, 2) to (2, 1) and maps (1, 3) to (-1, 3). Use the fact that

Answer 1

To find the linear transformation r: R² → R² that maps (5, 2) to (2, 1) and (1, 3) to (-1, 3), we can set up a system of equations and solve for the matrix that represents the linear transformation.

Step-by-step explanation:

A linear transformation is a function that maps vectors from one vector space to another in a way that preserves linear relationships. To find the linear transformation r: R² → R² that maps (5, 2) to (2, 1) and (1, 3) to (-1, 3), we can use the fact that the transformation is linear. Since we have two pairs of points, we can set up a system of equations and solve for the matrix that represents the linear transformation.

1. Let (x, y) be a point in R²
2. We can write the linear transformation as r(x, y) = (ax + by, cx + dy) where a, b, c, and d are constants to be determined
3. Using the given points, we have the following system of equations:
   • 5a + 2b = 2
   • 5c + 2d = 1
   • a + 3b = -1
   • c + 3d = 3
4. Solving this system of equations, we find that a = -1, b = 1, c = 1, and d = -1

Therefore, the linear transformation r: R² → R² is given by r(x, y) = (-x + y, x - y)