While doing some reading on projective geometry, I found the following proposition and proof about projective transformations here.
Proposition $7.2.4$. Projective transformations preserve the degree of curves.
Proof. Projective transformations map a monomial $X^i Y^i Z^k$ of degree $m=i+$ $j+k$ either to $0$ or to another homogeneous polynomial of degree $m$. If $f(X, Y, Z)$ is transformed by some transformation $T$ into the zero polynomial, then the inverse transformation maps the zero polynomial into $f$, which is nonsense.
I really don't understand how this proof is working. Isn't the very first sentence assuming the proposition is true already? And how is the inverse map "nonsense?"
What the author is pointing out boils down to the fact that when a projective transformation given by the matrix $A$ acts on the coordinates, it does so as follows: \begin{align*} [X:Y:Z] &= [U:V:W]A^{-1}\\ &= [U:V:W]\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix} \\ &= [a_{11}U+ a_{12}V+ a_{13}W: a_{21}U+ a_{22}V+ a_{23}W: a_{31}U+ a_{32}V+ a_{33}W] \end{align*} In particular, it takes each monomial $X^iY^jZ^k$ to the polynomial \begin{align*} X^iY^jZ^k &= (a_{11}U+ a_{12}V+ a_{13}W)^i(a_{21}U+ a_{22}V+ a_{23}W)^j(a_{31}U+ a_{32}V+ a_{33}W)^k \end{align*} which will also be a homogeneous polynomial of multidegree $m=i+j+k$ on formal variables $U, V, W$ (the degree zero reflects that a term could be mapped to a constant). To see this, use the trinomial expansion and let the algebraic dust settle.
The last bit handles the case where each monomial is mapped to a constant which must be zero since we are dealing with homogeneous polynomials. If $f$ is our original polynomial that is transformed into $A(f)$ by $A$ then the above description of the action on the coordinates tells us that $A^{-1}(A(f))$ will be the zero polynomial. Nonetheless, $A^{-1}(A(f))=f$ and we assumed $f$ was non-zero implicitly.
Let me know if it would be helpful to expand on this answer or if this level of detail is sufficient.