I am considering the Plücker map \begin{align*} \Phi:\mathbb{G}_k(\mathbb{C}^N)&\longrightarrow\mathbb{P}(\bigwedge^{\quad k}\mathbb{C}^N)\\ \langle z\rangle &\longmapsto\langle z_1\wedge\cdots\wedge z_k\rangle \end{align*}
My main goal is to prove that $\Phi$ is a closed embedding. Up to now I have been able to show all of the necessary criteria apart from the claim, $\Phi$ is an immersion.
I was given the definition:
Let $f:X\longrightarrow Y$ be a holomorphic mapping between complex manifolds $x\in X$ , $y\in Y$, then $f$ is called an immersion in $x\in X$ $: \iff f_*(x): \mathcal{T}_x(x) \longrightarrow > \mathcal{T}_y(f(x))$ is injective.
However, quite frankly I am stumped on how to show this.
Any thoughts or assistance would be greatly appreciated. Thanks in advance!