I'm having some issues understanding the pullback of a linear transformation. Specifically I'm doing some exercises from a differential geometry book of Andrew McInerney.
The question is
$G_0$ is the standard inner product and $S$ takes the determinant of two vectors in $\mathbb{R}^2$. They ask to calculate $T^*G_0$ and $T^*S$. But since $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$, at composing the two functions, $G_0$ would take values from $\mathbb{R}^2$ and $S$ values from $=\mathbb{R}^2\times \mathbb{R}^2$, this causes me confusion, because according to the definition $T^*S=S(T)$ and that would be undefined.