Problem 1: Let P2 be the space of polynomials in z of degree at most 2. Define T: P2 → P₂ by T(a + bz + cz²) = a+b + (b+2c)z + cz². • Find the matrix A of T with respect to the natural basis (1, z, z²) of P2. • Show that 1 is an eigenvalue of A of multiplicity 3. • Define E; = nullspace (A - I) for j = 1, 2, 3, and find a vector x in E3 which does not belong to E₂. • Define B' to be the basis of P2 which consists of elements whose coordinate vectors are (A - 1)²x, (A-1)x, x. . Find the matrix of T with respect to B'.

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