T: IR2 -> IR2, T(ax² + bx + c) = 4c + (2c - 3b)x + (2x + b + c)x² . What are the eigenvalues and eigenvectors of T?

Answer 1

T(ax² + bx + c) = 4c + (2c - 3b)x + (2a + b + c)x².

Let found images of standard basis vectors 1, x, x²

T(1) = 4·1 + 2x + x² = 4 + 2x + 2x²;

T(x) = - 3x + x²;

T(x²) = 2x². Because matrix A of linear transformation [(4 2 2) (0 -3 1) (0 0 2)]. Let find eigen-vectors of A:

4x + 2y + 2z = 0

- 3y + z = 0

2z = 0

From here z = 0, y = 0 and x = 0.

So null space for matrix A contain only zero-vector.