Past exam of the linear algebra course of the University of Oxford / 2023 / 1 / b / i / Solution

The dual space is the vector space of all linear functionals on $V$:Addition and scalar multiplication are defined pointwise. The dual map sends an output measurement back along $T$:The function composition of these linear maps is linear, and $T^{\prime }$ is linear in $\phi$. This is the algebraic dual map, so its definition involves no complex conjugation or inner product.