Talk:Generator preimage: Difference between revisions

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: Here's how it works:

:* We want to find u such that V*u = [0,..0,1,0,..,0] ~~for~~ the i-th index

:* We want to find u such that V*u = [0, ..., 0, 1, 0, ..., 0] where the 1 is on the i-th index.

:* Let's split up the problem:

:** V[i]*u = 1

:** V[no_i]*u = [0,0,...,0] (where V[no i] is V with the i-th row deleted)

:** V[no_i]*u = [0, 0, ..., 0] (where V[no i] is V with the i-th row deleted)

:* First calculate S. this is a kernel/comma basis for V[no_i].

:* The vector we want to find is in the rowspace of S. (aka it is a linear combination of its rows.) This is because ~~V*u = [0,..0,1,0,..,0] so~~ V[no i]*u = [0, 0, ... ,0] <~~br>~~=> u is in the kernel of V[no_i].

:* The vector we want to find is in the rowspace of S. (aka it is a linear combination of its rows.) This is because V[no i]*u = [0, 0, ... , 0] <=> u is in the kernel of V[no_i].

:* By calculating R = V[i] * S, we are basically calculating the number of steps V[i] maps each comma of S to.

:* build the block matrix [R | S] (prepending R to S)

:* if we do some row operations or [R | S], it stays valid. say we multiply it by an arbitrary unimodular matrix U: [R' | S'] = U*[R | S] = [U*R | U*S] this is true because R = V[i]*S, so U*R = U*V[i]*S = V[i]*U*S => R' = V[i]*S'

:* since we want V[i]*u = 1, we want to find some U so that R' = [1,0,..,0], this is exactly what the HNF does

: I have left out some details and matrix transposes. Hopefully this helps clear things up.

:-[[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 00:51, 18 December 2021 (UTC)

@@ Line 10: / Line 10: @@
 : Here's how it works:
-:* We want to find u such that V*u = [0,..0,1,0,..,0] for the i-th index
+:* We want to find u such that V*u = [0, ..., 0, 1, 0, ..., 0] where the 1 is on the i-th index.
 :* Let's split up the problem:
 :** V[i]*u = 1
-:** V[no_i]*u = [0,0,...,0]   (where V[no i] is V with the i-th row deleted)
+:** V[no_i]*u = [0, 0, ..., 0]   (where V[no i] is V with the i-th row deleted)
 :* First calculate S. this is a kernel/comma basis for V[no_i].
-:* The vector we want to find is in the rowspace of S. (aka it is a linear combination of its rows.) <br>This is because V*u = [0,..0,1,0,..,0] so V[no i]*u = [0, 0, ... ,0] <br>=> u is in the kernel of V[no_i].
+:* The vector we want to find is in the rowspace of S. (aka it is a linear combination of its rows.) <br>This is because V[no i]*u = [0, 0, ... , 0] <=> u is in the kernel of V[no_i].
 :* By calculating R = V[i] * S, we are basically calculating the number of steps V[i] maps each comma of S to.
 :* build the block matrix [R | S] (prepending R to S)
 :* if we do some row operations or [R | S], it stays valid. say we multiply it by an arbitrary unimodular matrix U:<br>[R' | S'] = U*[R | S] = [U*R | U*S]<br>this is true because R = V[i]*S, so U*R = U*V[i]*S = V[i]*U*S<br>=> R' = V[i]*S'<br>
 :* since we want V[i]*u = 1, we want to find some U so that R' = [1,0,..,0], this is exactly what the HNF does
+: I have left out some details and matrix transposes. Hopefully this helps clear things up.
 :-[[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 00:51, 18 December 2021 (UTC)