Let $\alpha: \mathbb R^n\to \mathbb R^k$ be an onto linear map. Let $\beta: \mathbb R^n \to \mathbb R^l$ be a linear map such that $(\alpha,\beta):\mathbb R^n\to \mathbb R^k\times \mathbb R^l $ is injective. Then there are $n-k$ coordinates in $\mathbb R^l$ such that $(\alpha,\beta_1\dots,\beta_{n-k})$ becomes a bijection.
My try: I know $(\alpha,\beta)$ is injective, therefore the columns of the representing matrix are independent. On the other hand, by surjectivity of $\alpha$, for every basis $B$, $\alpha(B)$ generates $\mathbb R^k$, and clearly $k\leq |\alpha(B)|\leq n$. But I can not make any reasoning to say there are $n-k$ coordinates in $ \mathbb R^l$ such that $(\alpha,\beta_1\dots,\beta_{n-k})$ becomes a bijection.