Let $n\ge 2$ and
$\theta\colon Spin(n)\rightarrow SO(n,\mathbb{R})$ be the two-fold covering of $SO(n,\mathbb{R})$ by the spin group $Spin(n)$, $\tilde{\theta}\colon \widetilde{GL^+}(n,\mathbb{R})\rightarrow GL^+(n,\mathbb{R})$ the connected two-fold covering.
Both $\theta$ and $\tilde{\theta}$ are homomorphisms of lie groups.
Using the theory of covering maps, one can show that there exists a homomorphism of lie groups
$j\colon Spin(n)\rightarrow \widetilde{GL^+}(n,\mathbb{R})$
such that $\tilde{\theta}\circ j= i \circ \theta$ where $i\colon SO(n,\mathbb{R})\rightarrow GL^+(n,\mathbb{R})$ is the inclusion.
$\textbf{Question:}$ Is $j$ injective? If yes, how can this be shown?