Invertible $T$ on $\mathbb F_2^d$ with $I+T$ invertible exists iff $d\ne1$ (Erdős 624 cross-review)

For a natural number $d$ there is a linear map $T:\mathbb F_2^d\to\mathbb F_2^d$ such that both $T$ and $I+T$ are bijective if and only if $d\neq1$. (For $d=0$ the space is trivial; for $d=1$ the only invertible map is $T=I$, and then $I+T=0$.)

This is the linear ingredient of the cross-review's local counterexample (blocks $E^1=\{p,q\}\cup\mathrm{supp}(z)$, $E^2=\{q,r\}\cup\mathrm{supp}(Tz)$, $E^3=\{p,r\}\cup\mathrm{supp}((I+T)z)$), used there with $d=k-6\ge58$.

Context: Erdős Problem 624 asks whether $H(n)-\log_2 n\to\infty$, where $H(n)$ is the least $m$ such that some $f:\mathcal P(X)\to X$ with $|X|=n$ satisfies $f[\mathcal P(Y)]=X$ for all $|Y|\ge m$. This is an auxiliary finite statement from Sol's Round 12 Step 17 cross-review / resilience analysis of the two-track (PAIR) case. It does not settle Erdős Problem 624, PAIR or Step 17.