Picked up a book on General Relativity for Mathematicians, but I'm a bit unclear on some of the tensor notation. For example, the Minkowski Metric
$$\eta_{\mu \nu} (\Delta x^\mu)(\Delta x^\nu)$$
Where $\eta_{\mu \nu}$ is the standard $4 \times 4$ identity matrix with an $-1$ in the (1,1) place.
I'm a bit confused as to how this notation works. Are $x^\mu$ and $x^\nu$ $4 \times 1$ matrices? (i.e. $\in \mathbb R^4$) If so, how does the product make sense? How is it then summed over to get
$$\eta_{\mu \nu} (\Delta x^\mu)(\Delta x^\nu) = -(\Delta x^0)^2 + \sum_{i=1}^{3} (\Delta x^i)^2?$$
Moreover, given a linear transformation $\Lambda$, how are
$$\Lambda^{\mu'}_\nu x^\nu$$ and $$x' = \Lambda x$$ the same?
Reference: http://arxiv.org/pdf/gr-qc/9712019.pdf
The book you are using employs the Einstein summation convention. This means that something of the form $f_{\mu}g^{\mu}$ is actually the same as $\sum\limits_{\mu=0}^{3}f_{\mu}g^{\mu}$. So, in $\eta_{\mu \nu} (\Delta x^\mu)(\Delta x^\nu)$, everything is being written as a scalar. In the Minkowski metric, this yields the summation you have written above.
For the second part, writing $x'=\Lambda x$ (regular matrix times vector multiplication) out component-wise yields $$x'=\Lambda x=\sum_{\nu} \Lambda^{\mu'}_{\nu}x^{\nu}=\Lambda^{\mu'}_\nu x^\nu$$