Let $G$ be a finite group, $p$ a fixed prime and $B$ be a $p-block$ of $G.$
Lemma 15.28 of Isaacs' Character Theory of Finite Groups says that each $\varphi\in IBr(G)\bigcap B$ can be written as a $\mathbb{Z}$ linear combination of Brauer characters of the form $\hat{\chi}$ for $\chi\in Irr(G)\bigcap B.$
Question: Can every $\hat{\chi}$ $\bigg(\text{where }\chi\in Irr(G)\bigcap B\bigg)$ be written as a linear combination of (probably a $\mathbb{C}$ linear combination?) of $\varphi\in IBr(G)\bigcap B$?
Thank you!!