Any proper subspace of $\mathbb{C}^{n}$ has the following property? Let $V$ be a proper linear subspace of $\mathbb{C}^{n}$ (viewed as a vector space over $\mathbb{C}$). I want to show that, then there exists $\alpha_{1},\dots,\alpha_{n}\in\mathbb{C}$ (not all zero), such that, for all $(\lambda_{1},\dots,\lambda_{n})\in V$,
$$
\alpha_{1}\lambda_{1}+\dots+\alpha_{n}\lambda_{n}=0.
$$
I suppose that, If a take the statement to be false, I can arrive somehow to the fact that $V=\mathbb{C}^{n}$, so if I suppose that, for all n-tuple of complex numbers not all zero ($\alpha_{1},\dots,\alpha_{n}\in\mathbb{C}$), we have that there exists an element $(\lambda_{1},\dots,\lambda_{n})$ in $V$ such that $\alpha_{1}\lambda_{1}+\dots+\alpha_{n}\lambda_{n}\neq 0$, I could take $v\in\mathbb{C}^{n}$ and prove that $v\in V$ (by using the previous hypothesis). Nevertheless, I don't know if this would be the right way to think it, and even if it is, I am stuck on the proof completely...
Can someone provide just some guidance so I can prove it ? Thanks in advanced!
 A: For $a,b\in\mathbb C^n$, with $a=(a_1,a_2,\ldots,a_n)$ and $b=(b_1,b_2,\ldots,b_n)$, define $a\cdot b:=a_1b_1+a_2b_2+\ldots+a_nb_n\in\mathbb C$. It is easy to see that the map $\cdot:\mathbb C^n\times\mathbb C^n\to\mathbb C$ is bilinear.
Now, let $(v_1,v_2,\ldots,v_k)$ be a basis of $V$, where $k<n$ i.e. $k\le n-1$.
Further, define a map $L:\mathbb C^n\to \mathbb C^k$ given by $L(a):=(a\cdot v_1,a\cdot v_2,\ldots,a\cdot v_k)$ for $a\in\mathbb C^n$. It is easy to prove that $L$ is linear. As per Rank-Nullity theorem, $\dim\ker(L)=n-\dim\operatorname{im}(L)$. As $\operatorname{im}(L)\subset \mathbb C^k$, we have $\dim\operatorname{im}(L)\le k\le n-1$. Thus, $\dim\ker(L)\ge n-(n-1)=1$. So, $\ker L$ is nontrivial, i.e. you can find $a\in\ker L, a\ne 0$.
However, this means $a\cdot v_1=a\cdot v_2=\ldots=a\cdot v_k=0$, which then implies that $a\cdot v=0$ for all $v\in V$.
Note that the same proof applies to finite-dimensional vector spaces over any field, not necessarily $\mathbb C$.
A: Hint: Let $e_1,e_2,\dots,e_k$ be an orthonormal basis for $V$. Take any $w$ in the complement of $V$.
Analyze the vector
$$ w^{'} = w - \sum_{i=1}^{k}  \langle w,e_i \rangle e_i $$
(an $n$ tuple of complex numbers)
Note that for $a,b \in \Bbb C^n$,
$$ \langle a, b \rangle  =a_1 \bar{b_1}+a_2 \bar{b_2}+\ldots +a_n \bar{b_n}$$
