I need help understanding the solution. Here it is:
Solution. Starting from the last row, one has $x_{N} = y_{N}/A_{NN}$, which requires one division. At row number $i$, one has $x_{i} = (y_{i} - \sum_{j=i+1}^{N} A_{ij} x_{j})/A_{ii}$, which is one division and $N - i$ multiplications. This gives $N$ divisions and $\sum_{i=1}^{N} (N - i) = \sum_{k=0}^{N - 1} k N(N - 1)/2$ multiplications for a total of $N^{2}/2 + N/2$
First of all, I don't really get their notation. What is $x_{N}$ and $y_{N}$? Why are they dividing? Can't they just multiply $Ax$ to get $y$? I got that this requires $1N + 2N + 3N + \cdots + N \cdot N$ total steps.
Can someone please clarify?
Thanks