grandes-ecoles 2013 Q3

grandes-ecoles · France · x-ens-maths1__mp Sequences and Series Recurrence Relations and Sequence Properties
For $i \in \mathbf{Z}$, we define $v_i$ in $\mathbf{C}^{\mathbf{Z}}$ by $$v_i(k) = \left\{\begin{array}{l} 1 \text{ if } k = i \\ 0 \text{ if } k \neq i \end{array}\right.$$
3a. Show that the family $\{v_i\}_{i \in \mathbf{Z}}$ is a basis of $V$.
3b. Calculate $E(v_i)$. 
For $i \in \mathbf{Z}$, we define $v_i$ in $\mathbf{C}^{\mathbf{Z}}$ by
$$v_i(k) = \left\{\begin{array}{l}
1 \text{ if } k = i \\
0 \text{ if } k \neq i
\end{array}\right.$$

3a. Show that the family $\{v_i\}_{i \in \mathbf{Z}}$ is a basis of $V$.

3b. Calculate $E(v_i)$.
Paper Questions
Q1a Q1b Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23