Inverse of a Matrix of Partial Derivatives

Question

This is from the book Lectures on Finsler Geometry by Zhongmin Shen.I am stucked here to find inverse of $a_{ij}$ that is $a^{ij}$ and how to prove that $||\beta||<1$ so please help I have tried it many times but couldn’t do it.

Answer 1

Let $v_1,\dots,v_m$ be linearly independent vectors in $\Bbb R^{m+1}$, and let $B=(B_\mu)$ be a vector in their span. Let $a_{ij} = v_i\cdot v_j$, and let $(a^{ij})$ be the inverse matrix of $(a_{ij})$.

Write $B = \sum c_jv_j$. Then $b_i = B\cdot v_i = \sum a_{ij}c_j$, so we can solve for $c_j = \sum a^{ji}b_i$. It follows that $$\sum B_\mu^2 = B\cdot B = \sum a_{ij}c_ic_j = \sum a_{ij}a^{ik}a^{j\ell}b_kb_\ell=\sum\delta_i^\ell a^{ik}b_kb_\ell=\sum a^{\ell k}b_kb_\ell.$$ It would seem the author's inequality should be an equality.