Exercise 18, § 1

a) Show that in the vector space of maps from $\mathbf{R}$ into $\mathbf{C}$, the distinct functions of the form $x^n e^{\alpha x}$ ($n$ an integer, $\alpha$ arbitrary complex) form a free system (argue by contradiction, considering a relation among these functions having the least possible number of coefficients $\neq 0$, and differentiating this relation).

b) Let $f(X_1, X_2, \ldots, X_m)$ be a polynomial in $m$ indeterminates, with complex coefficients, such that when one substitutes the function $\cos \left( \sum_{k=1}^n p_{jk} x_k \right)$ for $X_j$ for $1 \leq j \leq r$, and the function $\sin \left( \sum_{k=1}^n p_{jk} x_k \right)$ for $r+1 \leq j \leq m$, where the $p_{jk}$ are real and the $x_k$ are real, one obtains an identically zero function of the $x_k$; show that the same identity holds when one gives the $x_k$ arbitrary complex values (use a)).