= Solution
Every ordered <monomial> in the degree-$n$ homogeneous component has the form $x^{n-j}y^j$, $0\le j\le n$. The formulas in part (a) show
$$
\begin{aligned}
K(x^{n-j}y^j)&=q^{n-2j}x^{n-j}y^j,\\
E(x^{n-j}y^j)&=[j]x^{n-j+1}y^{j-1},\\
F(x^{n-j}y^j)&=[n-j]x^{n-j-1}y^{j+1}.
\end{aligned}
$$
Each nonzero image still has total degree $n$; $K^{-1}$ does too. Since $E,F,K,K^{-1}$ generate the <quantum enveloping algebra of sl2>, \b[the degree-$n$ component is a <submodule>], with <basis> $x^n,x^{n-1}y,\ldots,y^n$ and dimension $n+1$.
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