Does anyone know of a good textbook on non-Eucledian geometry, which approaches geometry by using mostly linear algebra (e.g., projections, dot product, cross product, etc.)? I've looked at the "classical" book by Marvin Greenberg, but it doesn't appear to be using much linear algebra there.
One example of a spherical geometry problem which uses vectors (and thus linear algebra machinery is used for solution):
Let $u = \frac{1}{\sqrt{3}}(1, 1, -1)$ and $v = \frac{1}{\sqrt{2}}(1, 0, 1)$. Find the point $w\in \mathbb{S}^2$ such that $L_w$ is the line through $u$ and $v$. [Here $L_w$ is the spherical line with normal vector $w$].
I'd appreciate it very much if there's a person who knows of such a good textbook, or maybe even more than one.