Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 8 / c

Write the ordered basis as $\mathcal{B}=(1,t,e^t,t{e}^t)$. Differentiation givesThe corresponding coordinate vectors form the columns of the representing matrix:This is an example of differentiation on an exponential-polynomial space.

Choose $\boxed{v_4=e^t(1-t)}$, and denote the new ordered basis by $\mathcal{C}=(b_1,b_2,b_3,b_4)$. The change of basis matrix, with columns expressed in $\mathcal{B}$, isThis nonzero determinant proves that the proposed fourth function completes a basis. Express the derivatives in that basis:Reading off columns therefore gives

Yes. The old matrix is triangular, with determinant zero and trace $0+0+1+1=2$. The upper-left two-by-two block of the new matrix has determinant zero, so the whole determinant remains zero. Its trace isThusThis verifies both change of basis invariants from part (b).