Definition of linear chains

Question

I am reading Hatcher books in algebraic topology and he defined by $LC_n(Y)$ as the subgroup of $C_n(Y)$ generated by linear maps $\Delta^n$ $\to Y$ where $C_n(Y)$ is the abelian group of the $n$ singular chains and $Y$ is a convex subset of $R^n$ . I don't understand this definition because I don't know what is a linear map $\Delta^n$ $\to Y$ ? Is it just a map where its image is homeomorphic to $\Delta^n$? or maby the singular chains of $C_n(Y)$ that map to 1 dimensional simplices? I cannot find the definition anywhere..
Thanks in advance

Answer 1

Hatcher mentions that $Y$ is assumed to be a convex subset of some Euclidean space.

In that case, a continuous map $\sigma : \Delta^k \to Y$ is called linear if $\sigma(a_0v_0 + a_1v_1 + \cdots a_kv_k)$ $=$ $a_0\sigma(v_0) + \cdots a_k\sigma(v_k)$ where $\sum_i a_i = 1$ and $\Delta^k$ is the standard $k$-simplex embedded in $\Bbb R^k$.