Problem 1 Let Z denote the integers. Let S = {0 a,b e z} (a) Prove that S is a subring of M2 (Z) (b) Let I= {[ rse z. You can assume I is an additive subgroup of M2 (Z). Prove %3D that I is a two-sided ideal of S by checking the ideal condition on both sides.

T= Consider the subrings а C { [° d]|a,c,d € Q} C, 0 and U = { [88]|₁ € Q} of the matrix ring M₂(Q). Show that U is an ideal in the subring T but not an ideal in M₂(Q).

Let R be the field of real numbers and {e1, 2} be the standard basis Problem 5 for R2. Consider the linear operator TE L(R²) defined by r(e₁) = 3e₁ - e2, 7(₂)=4e1-202- Find the minimal polynomial for 7 and show that the rational canonical form for 7 is T R -1 What are the elementary divisors of r? Hint: Given TEL(V), if m,(x) = av a₁x +...+ak is the minimal polynomial for T th B = {U, T(U),...,7-¹(v)} is a basis for V and [T]B is the companion matrix of r and gir the minimal polynomial. Now, consider v = e₁

I have the solution to this question I just need 2-3 examples of this: