Definition. Let $\mathsf{V}$ and $\mathsf{W}$ be vector spaces, and let $\mathsf{T}:\mathsf{V}\to\mathsf{W}$ be linear. A function $\mathsf{U}:\mathsf{W}\to\mathsf{V}$ is said to be an inverse of $\mathsf{T}$ if $\mathsf{TU}=\mathsf{I_W}$ and $\mathsf{UT}=\mathsf{I_V}$. If $\mathsf T$ has an inverse, then $\mathsf T$ is said to be invertible. If $\mathsf T$ is invertible, then the inverse of $\mathsf T$ is unique and is denoted by $\mathsf T^{-1}$.
Given the definition of invertible and inverse of linear transformation above, my question is that is $\ $ "$\mathsf{T}:\mathsf{V}\to\mathsf{W}$ be linear" a necessary condition for a transformation, which may not be linear, being invertible? Or it's just because it's the only concern here (since it's linear algebra)?
It's just because the context here is linear algebra. For instance, $c\colon\mathbb{R}\longrightarrow\mathbb R$ defined by $c(x)=x^3$ is not linear, but it is invertible; its inverse is $x\mapsto\sqrt[3]x$.