## Spaces – First Round

Hi, I am back.

Things have progressed reasonably well. So, I am reasonably happy with my math skills now, though I didn’t fix all concepts. I am now crossing my fingers for no relapse.

For the time being, my definition of mathematics is recognizing patterns, abstracting them, and using the results in conjunction with previous abstractions. It remains to see whether this will survive.

Back to Space, the issue of Vector Space has undergone a good treatment recently, in Prof Gowers weblog. Do pay attention to Terence Tao‘s comment and his linked notes. I personally find the linear transformation approach more useful than matrices, to the extent that I sometimes think of a matrix entry as $a_{ij}=e_i^\prime A e_j$, just to avoid breaking the conceptual transformation approach. I probably find that easier as a result of using SciPy, where up to a certain extent you gain in performance by thinking of matrices as black boxes, akin to vectorization. Probably Matlab users will have the same effect. By the way, Tao’s comment about the idea of a linear transformation being a multidimensional generalisation of a ratio did not help at first, but it is now resonating with another idea that I have. More on that later. In short, the concept of linear transformation is that embedded in vector space.