Linear group, Gaussian elimination summary

1. Linear groups:
　　　Definition: linear set is a group of the original can be set or a different set of identical minimum set.
　　　Seeking: the main application of the following properties: if x is a linear element group, y is an element to be added, then x ^ y be a linear group.
　　　　　　Proof: set = Z \ (X \) ^ \ (Y \) , as long as the use of z, with the equivalent \ (X \) ^ \ (Y \) , to a y, can be used \ ( Z \) ^ \ (X \)
　　　Therefore, for each scanned binary number, if the i-th bit is 1, it is determined whether the hundreds, if so, the number or different, otherwise the number of bits on the i set this number, and the number of end scanning.
　　　The combined: the number of a group of linear force to insert another
　　　set number of length l, then n number of build complexity is linear group \ (O (NL) \) .
　　　The combined complexity \ (O (L ^ 2) \) .
　　　It does not support removing.
　　　Application: XOR and involves seeking the maximum / k XOR and large, a number can be determined whether an exclusive OR, etc.
　　　divergent or group and is a number of 0: first group to Linear, the answer is \ (^ 2 Total {- size linear yl} \)
　　　linearly independent set: available Gaussian elimination determined, each number is to be used other elimination, if not the elimination, it will be added.
　　　example:
　　　 P3292 [SCOI2016] lucky numbers
　　　 maximum XOR and asked path.
　　　 Pre-multiplying the linear group, and then combined to violence
　　　 [WC2011] XOR and the maximum path
　　　 seeking FIG. 1 ~ n XOR all paths and maximum.
　　　 Pretreatment of the linear ring dfs encountered, then free to find a path, and to seek the maximum XOR with the rings.
　　　 [HAOI2017] eight vertical and eight horizontal :
　　　 support edging, edge deletion, modification right side, and ask the maximum and XOR ring.
　　　 Xor and similar path, the exclusive OR result of the inquiry is the greatest and, using linear group all rings.
　　　 Modify delete + insert can be seen.
　　　 Since the linear base does not support removing, so use divide and conquer tree line.
　　　 With disjoint-set to maintain all of the rings.
2. Gaussian elimination:
　　　time complexity \ (O (n ^ 3)
\) 　　　Code:
　　　Note: When elimination, elimination coefficient seek out first.
　　　Application: matrix inversion, to a cycloalkyl probability DP, lights problem, like
　　　the exclusive OR Gaussian elimination can be used bitset.