Let $m\in\mathbb{N}$ and $k$ be an algebraically closed field. Consider $$X_1:=(x^1_1,\cdots,x^1_n), X_2:=(x^2_1,\cdots,x^2_n),\cdots,X_m:=(x^m_1,\cdots,x^m_n)$$ which are nonzero and whose components are in $k$. (nonzero means $X_i\neq(0,0,\cdots,0)$)
My question is:
Is there a $\Lambda:=(\lambda_1,\cdots,\lambda_n)\in k^n$ such that $X_i \circ \Lambda\neq 0$ (inner product) for every $i$?.
Given a nonzero vector $X$ in $k^n$, the set of all $\Lambda$ in $k^n$ such that $X\cdot\Lambda=0$ is a proper subspace of $k^n$. $k$, being algebraically closed, is an infinite field, and no (nontrivial) vector space over an infinite field can be expressed as a finite union of proper subspaces (there's a question about proving this, either here or on MathOverflow, maybe you can search for it if you're interested). So there must be $\Lambda$ in $k^n$ such that there is no $i$ for which $X_i\cdot\Lambda=0$.
