Linear System Problem Generator

First, make choices in the blue box on the left below to randomly populate a matrix in reduced row echelon form. Then make choices in the gray box labeled 'Operations' in the middle column. (Keyboard shortcuts are given below.) This will randomly create a matrix in the red box on the right below which is row equivalent to the matrix in the blue box. Below that, two problems along with their solutions are given. The first problem asks for the solution to the homogeneous linear system associated with the matrix in the red box, and the second problem asks for the solution to the nonhomogeneous linear system associated with the matrix.

The $\mathbb{Z}$-radius is the radius of the closed interval about the origin from which the associated integers will be randomly chosen. For example, if you choose '3' for the $\mathbb{Z}$-radius of a shear in the 'Operations' column, then an operation of the form $R_i \rightarrow R_i + c R_j$ will be performed on the matrix, where $c = -3, -2, -1, 1, 2,$ or $3$.

There are 6 latex strings that can be copied for use in a latex document.

1. The RREF in the blue box.
2. The row equivalent matrix in the red box.
3. The problem about the homogeneous linear system.
4. The solution to the problem about the homogeneous linear system.
5. The problem about the nonhomogeneous linear system.
6. The solution to the problem about the nonhomogeneous linear system.

The Shear By Pivot algorithm performs a large collection of shears all at once. The goal is to remove many of the zeroes in the pivot columns. For every pivot row, some nonzero multiple of that row (depending on the $\mathbb{Z}$-radius of the shear scalar) is added to the other rows.

Finally, the problems ask for the solutions to the systems to be given in 'Parametric Form'. For a description of parametric form, please see section 4 of the following linear algebra lesson about linear systems:

https://web.ma.utexas.edu/users/jmeth/DESummer19/LinAlg2.html

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