Let $V$ be a real infinite-dimensional vector space of cardinality $\kappa$. Does there exist a set $\Omega$ of cardinality $\kappa$ of linear maps from $V$ to $V$ such that for every $n\geq 1$, every nonzero vector $(x_1,\ldots, x_n) \in \mathbb{R}^n$, and distinct $A_1,\ldots, A_n \in \Omega$, the map $$x_1A_1+\cdots+x_nA_n$$ is an isomorphism from $V$ to $V$?
When $V$ is an infinite-dimensional separable Hilbert space, there exists a countable set $\Omega$ such that $$U^2=-I,~UV+VU=0,~~~~~~~~~~~~~~~(1)$$ for all distinct $U,V \in \Omega$. Then every nontrivial linear combination of elements in $\Omega$ is invertible. The question is that if generally one can forego the conditions (1) but increase the cardinality of the set and still have invertible nontrivial linear combinations.