Tverberg's theorem, a new proof

We give a new proof Tverberg’s famous theorem: For every set $X\subset {\mathbb{R}}^d$ with $\mathrm{\mid }X\mathrm{\mid }=(r-1)(d+1)+1$, there is a partition of $X$ into $r$ sets $X_1,\dots ,X_r$ such that ${\bigcap }_{p=1}^r\text{conv}{X}_p\mathrm{\ne }\mathrm{\varnothing }$. The new proof uses linear algebra, specially structured matrices, the theory of linear equations, and Tverberg’s original ``moving the points" method.