Let P denote the vector space of all polynomials and fx deno

Let P denote the vector space of all polynomials and f\'(x) denote the derivative of f Element P. State \"True\" or \"False\" for each of the following statements. You do not need to justify your answers. T: R^2 rightarrow R^3 defined by T(x, y) = (pi^2 x, 0, y/2) is a linear transformation. T: R^2 rightarrow R^3 defined by T(x, y) = (x+y, x - y, xy) is a linear transformation. T: R^2 rightarrow R^3 defined by T(x, y) = (1, 0, 0) is a linear trans-formation. T: R^2 rightarrow R^3 defined by T(x, y) = (y, x, y) is a linear trans-formation. T: R^2 rightarrow R^3 defined by T(x, y) = (x, x^2, x^3) is a linear transformation. T: R^2 rightarrow R^3 defined by T(x, y) = (2x + 3y, 3x +4y, 4x + 5y) is a linear transformation. T: P rightarrow P defined by T(f(x)) = f(0) + f\'(0)x + f\"(0)x^2 is a linear transformation. T: P rightarrow P defined by T(f(x)) = f(1) + f\'(2)x + f\"(3)x^2 is a linear transformation. T: P rightarrow P defined by T(f(x)) = f(x - 3) is a linear trans-formation. T: P rightarrow P defined by T(f(x)) = f(x) - 3 is a linear trans-formation.

Solution

1) True
2)False
3)True

4)True
5)False
6)True
7)True

8)False

9)False

10)False