UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 1

Transformer för beräkningar 2001 03 21 01trp1

1.1. Calculate (1 + i\sqrt{3})^{6} (a) using polar representation; (b) using the binomial formula... if you do not get too tired, that is.

1.2. Solve completely the equation z^{7} = 7, z ∈ C.

1.3. Find the dimension of the space of all polynomials of one complex variable which are of degree at most 17 and vanish at the points z = 0, z = 1, z = 2.

1.4. Is the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) diagonizable?

1.5. Let V be the space of all polynomials of one real variable of degree at most m. Is the mapping T: V ∋ f ↦ f′ ∈ V injective? Is it surjective? Determine the dimensions of im T and ker T. Determine the eigenvalues of T as well as all eigenvectors. Is T diagonizable?

1.6. We compare convolution on Z and on Z_{N} = Z/NZ, N ⩾ 2. Let a = δ_{−1} − δ_{0}, where we define δ_{k}(j) to be zero for j ≠ k and to be one for j = k. Prove that on Z the equation a ∗ z = b has a solution for any given b, and that the space of solutions to a ∗ z = 0 has dimension 1. By way of contrast, show that on Z_{N}, the equation a ∗ z = b can be solved only if b satisfies a certain condition—and find this condition. For the homogeneous equation, the situation for Z_{N} is the same as for Z. (The lesson is that, on Z, the equation a ∗ z = b with periodic b can have nonperiodic solutions z.)

1.7. Define a mapping T: l^{2}(Z_{4}) → l^{2}(Z_{4}) by

T (z) = (2z(0) − z(1), iz(1) + 2z(2), z(1), 0).

Is T linear? Is T translation invariant? Calculate T (R_{1}z) and R_{1}(T z) for z = (1, 0, −2, i)^{T}.

1.8. Let S, T be two translation-invariant linear mappings of l^{2}(Z_{N}) into itself. Prove that they commute, i.e., that T ∘ S = S ∘ T. Deduce that circulant matrices always commute—try also to prove this fact by a direct calculation using the definition of circulant matrices.

1.9. Prove that T (z ∗ w) = T (z) ∗ w = z ∗ T (w) if T is a translation-invariant linear mapping of l^{2}(Z_{N}) into itself.

1.10. Prove that a linear mapping T of l^{2}(Z_{N}) into itself is translation invariant if and only if it is a polynomial in the translation operator R_{1}.