Finding Matrix Representation for Identity Transformation

Note that Linear Maps Determined by Their Actions on A Basis so, calculate: $I(v_1)=v_1=1\ast v_1+0\ast v_2$ $I(v_2)=v_2=0\ast v_1+1\ast v_2$ This gives a matrix:

$$[T{]}_{\alpha }^{\alpha }=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

Note that $I(v_1)$ occupies the first column of the matrix Note that $I(v_2)$ occupies the second column of the matrix