Show that $T_H$ is a linear subspace of $T_G$ so that $\dim H\leq \dim G$ 
Let $H$ be a subgroup of a matrix group $G$.
  Show that $T_H$ is a linear subspace of $T_G$ so that $\dim H\leq \dim G$

Definition:
Let $\phi:G\rightarrow H$ be a smooth homomorphism of matrix groups.
If $\gamma '(0)$ is a tangent vector to $G$ at $I$, we define a tangent vector $d_\phi(\gamma '(0))$ to $H$ at $I$ by $$d_\phi(\gamma '(0))=(\phi\circ\gamma)'(0)$$ The resulting map $d_\varphi:T_G\rightarrow T_H$  is called the differential of $\phi$.
To see if a vector space $V$ is a subspace of some other (higher-dimension) vector space, we need to check the following:


*

*$V\neq\emptyset$

*$\vec{x},\vec{y}\in V \implies \vec{x}+\vec{y}\in V$

*$\alpha\in \mathbb F, \vec{x}\in V\implies \alpha\vec{x}\in V$
But I don't know how to apply these to the question.
Any help is appreciated

It looks like your book is defining tangent vectors as derivaties of somoth curves. But if $\gamma$ is a curve landing in $H$ then a priori it lands in $G$, so the injectivity is automatic.
