Let $T : \mathbb R^n \rightarrow \mathbb R^m$ is a Linear Transformation. Prove that $\operatorname {im}(T)$ is a subspace of $\Bbb R^m$

Question

Let $T : \mathbb R^n \rightarrow \mathbb R^m$ is a Linear Transformation. Prove that $im(T)$ is a subspace of $R^m$

I am very bad at problems like this, I know what defines a subspace, I'm just uncertain on how to apply it.

Answer 1

$T(0)=0$ is true since $T(0)=T(0+0)=2T(0)$ and for any $y_1,y_2\in \text{im}(T)$, there exists $x_1, x_2$ such that $T(x_i)=y_i$ ($i=1,2$). So $y_1+y_2=T(x_1)+T(x_2)=T(x_1+x_2)$ so that $y_1+y_2\in \text{im}(T)$.

Closure under scalar multiplication is proved similarly.