In the example where Y is sampled twice and we have the equations 1s1 + 0s2 + 1s3 = 0 and 1s1 + 1s2 + 1s3 = 0, what are the solutions for s1, s2, and s3?
In the context of the Simon's algorithm, let's consider the given equations: 1s1 + 0s2 + 1s3 = 0 and 1s1 + 1s2 + 1s3 = 0, where s1, s2, and s3 are unknown variables. These equations represent a system of linear equations, and we need to find the solutions for s1, s2, and s3.
To solve this system of equations, we can use various methods such as Gaussian elimination, matrix inversion, or substitution. Let's proceed with the Gaussian elimination method to find the solutions.
First, let's write the given system of equations in matrix form:
|1 0 1| |s1| |0|
|1 1 1| * |s2| = |0|
|s3|
Now, we will perform Gaussian elimination on this matrix. The goal is to transform the matrix into the reduced row-echelon form. Let's start with the first column:
1 0 1 0
1 1 1 0
Subtracting the first row from the second row, we get:
1 0 1 0
0 1 0 0
Now, the system of equations can be written as:
s1 + s3 = 0
s2 = 0
From the second equation, we can conclude that s2 = 0. Substituting this value into the first equation, we get s1 + s3 = 0. This implies that s1 = -s3.
Therefore, the solutions for s1, s2, and s3 are: s1 = -s3, s2 = 0, and s3 can take any value.
The given system of equations has infinitely many solutions, where s1 = -s3, s2 = 0, and s3 can be any real number.
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