[PROOFS] Geometric examples

For $x,y\in\mathbb R^n$ and $\lambda\in\mathbb R$ ,

T_A(x+y)=A(x+y)=Ax+Ay=T_A(x)+T_A(y),

and

T_A(\lambda x)=A(\lambda x)=\lambda Ax=\lambda T_A(x).

Thus $T_A$ satisfies both conditions in the definition of a linear map.

Write $x=x_1e_1+\cdots+x_ne_n$ . Preservation of finite linear combinations gives

T(x)=x_1T(e_1)+\cdots+x_nT(e_n).

These vectors are exactly the columns of $A$ , so the right-hand side is $Ax$ .

Proofs from Geometric examples