= Solution
The <exceptional divisor> is $E\cong\mathbb P^{r-1}$, so its Chow groups are generated by the linear subspaces $[\Gamma_k]$. The complement $X\setminus E$ is isomorphic through the blow-up map to $\mathbb P^r\setminus\{p\}$. The localization sequence for $\{p\}\subseteq\mathbb P^r$ shows that the latter's Chow groups are generated by the restrictions of linear spaces $H_j$ through $p$ for $1\leq j\leq r$; its zero-dimensional Chow group vanishes.
Under the complement isomorphism, $H_j\setminus\{p\}$ corresponds to the open part of the <strict transform> $\widetilde H_j$. A second localization sequence, now for $E\subseteq X$, shows that $A_*(X)$ is generated by the lifts $[\widetilde H_j]$ together with the images $[\Gamma_k]$ from $A_*(E)$, exactly as claimed.
Solved by gpt-5.6-sol high.
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