But paragraph which I marked by red line seems to me confusing.
Let $k=3$ then $\sigma=[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3]$ and $\partial \sigma=[\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3]-[\mathbf{p}_0,\mathbf{p}_2,\mathbf{p}_3]+[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_3]-[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_2].$ Since $T:E\to V$ is affine then $T(\mathbf{x})=T(a)+A\mathbf{x}$ for some $a\in E$ and $A\in L(\mathbb{R}^n,\mathbb{R}^m)$.
1) We will show that $T\circ \sigma$ is oriented affine $k$-simplex. $$T\circ \sigma(u)=T(a)+A(\mathbf{p}_0+u_1(\mathbf{p}_1-\mathbf{p}_0)+u_2(\mathbf{p}_2-\mathbf{p}_0)+u_3(\mathbf{p}_3-\mathbf{p}_0))=$$$$=[T(a)+A\mathbf{p}_0,T(a)+A\mathbf{p}_1,T(a)+A\mathbf{p}_2,T(a)+A\mathbf{p}_3];$$
2) We have to show that $\partial (T\circ \sigma)=T(\partial \sigma)$.
$$\partial (T\circ \sigma)=[T(a)+A\mathbf{p}_1,T(a)+A\mathbf{p}_2,T(a)+A\mathbf{p}_3]-[T(a)+A\mathbf{p}_0,T(a)+A\mathbf{p}_2,T(a)+A\mathbf{p}_3]+[T(a)+A\mathbf{p}_0,T(a)+A\mathbf{p}_1,T(a)+A\mathbf{p}_3]-[T(a)+A\mathbf{p}_0,T(a)+A\mathbf{p}_1,T(a)+A\mathbf{p}_2]$$ by definition 10.29.
But $$T(\partial \sigma)=T(a)+A(\mathbf{p_1}+u_1(\mathbf{p_2-\mathbf{p_1}})+u_2(\mathbf{p_3}-\mathbf{p_1}))-A(\mathbf{p_0}+u_1(\mathbf{p_2-\mathbf{p_0}})+u_2(\mathbf{p_3}-\mathbf{p_0}))+A(\mathbf{p_0}+u_1(\mathbf{p_1-\mathbf{p_0}})+u_2(\mathbf{p_3}-\mathbf{p_0}))-A(\mathbf{p_0}+u_1(\mathbf{p_1-\mathbf{p_0}})+u_2(\mathbf{p_2}-\mathbf{p_0}))=T(a)+\partial(T\circ \sigma)$$ and we see that $T(\partial \sigma)\neq\partial(T\circ \sigma)$.
Can anyone please explain what's wrong with my reasoning?