LetFbe the linear transformation that switches $\displaystyle \vec e_1 + \vec 2e_2$ for $\displaystyle \vec 2e_1 + \vec e_2$ and vice versa. Find the transformation matrixAfor this linear transformation.

Then find the vectors $\displaystyle \vec f_1, \vec f_2$ so that $\displaystyle F(\vec f_1)=\vec f_1$ and $\displaystyle F(\vec f_2)=-\vec f_2$. Choose this as a basis, and then find F's transformation matrix in this basis.