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evinda

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Let $F$ be a field with infinite elements and $V$ a $F$-linear space of dimension $n$ and $W_1, \dots, W_m$ subspaces of $V$ of dimension $n_i<n, i=1, \dots, m$. We want to show that $V \setminus{(W_1 \cup \cdots \cup W_m)} \neq \varnothing$.

- Fix a basis $\{ v_1, \dots, v_n\}$ of $V$. Show that there is a non-zero linear mapping $\ell_k: V \to F$, such that $W_k \subset ker(\ell_k)$ (i.e. $w \in W \Rightarrow \ell_k(w)=0$).
- Construct a non-zero polynomial $f_k(X_1, \dots, X_n) \in F[X_1, \dots, X_n]$ such that $x_1 v_1+\dots+ x_n v_n \in W_k \Rightarrow f_k(x_1, \dots, x_k)=0$.
- Consider as given that if $f \in F[X_1, \dots, X_n]$ is a non-zero polynomial, then there is a point $(a_1, \dots, a_n) \in F^n$ such that $f(a_1, \dots, a_n) \neq 0$. Show that there is a vector $v \in V \setminus (W_1 \cup \dots \cup W_m)$.

Could you help me to find the desired linear mapping $\ell_k$ ? (Thinking)