On page 7 of Milnor's Morse Theory is part of a proof of the Morse Lemma:
Suppose by induction that there exist coordinates $u_1, \ldots, u_n$ in a neighbourhood $U_1$ of $0$ so that $f = \pm(u_1)^2 \pm \cdots \pm (u_{r-1})^2 + \sum_{i,j \geq r} u_i u_j H_{ij}(u_1,\ldots,u_n)$ throughout $U_1$; where the matrices $(H_{ij}(u_1,\ldots,u_n))$ are symmetric. After a linear change in the last $n - r + 1$ coordinates, we may assume that $H_{rr}(0) \neq 0$.
(full text of proof: page 6, page 7, page 8)
I do not understand how he can WLOG assume that $H_{rr}\neq 0$ by doing a "linear change of variables". I don't really even understand what he's saying. Could someone please spell this out for me? Note that page 6 handles an entirely different part of the lemma. If you need context, it should be enough to just assume that the equalities at the top of page 7 hold. You also must know that $f$ has a nondegenerate critical point at $0$, and $f(0)=0$.