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HINGES - An Illustration of Gauss-Jordan Reduction - Outcomes and Solution

Przemyslaw Bogacki

I designed the first stage to reveal initially relatively little information to get students to begin thinking about the problem. In subsequent stages I gradually provided additional clues -- both visual and analytic -- to help students succeed. I told the class that I would award full marks for the correct solution of the problem regardless of the stage during which the solution was submitted.

While the entire class of 35 students accessed the stage 1 "leaflet" document, they appeared to have difficulty solving the puzzle in the "virtual" mode. Only a handful of students submitted their solutions at that stage, and none of them were correct.

Sixteen students executed the sequence of moves in stage 2, almost all of them providing the correct frame number values, as expected. Three students submitted correct solutions, linking the puzzle to the Gauss-Jordan reduction and elementary row operations. Three additional students made the correct connection after studying the document in stage 3.

Having just six students out of 35 successfully complete this activity may not strike someone as a particularly impressive outcome. However, I viewed the experiment described here as a "proof of concept", and, in this sense, it has been successful. One can modify the activity described here to make its appeal broader, by providing additional information to students, increasing the credit value, etc.

At the conclusion of the activity, I posted the solution:

Every time we turn a door plane to make it parallel to one of the axes (x,y, or z), we end up removing the corresponding term from that equation (e.g., the plane 2y+3z=5 is parallel to the x axis). Therefore, each move corresponds to an elementary row operation, and the entire game corresponds to the Gauss-Jordan reduction leading to the reduced row echelon form.

I illustrated this by the animated sequence of correct HINGES moves along with the corresponding elementary row operations.

Here are a few additional points, which you may want to discuss with students after they participate in the HINGES activity:

  • It can be shown that the combined equation
    (pa1+qa2)x + (pb1+qb2)y +  (pc1+qc2)z = pd1+qd2,
    resulting from taking a linear combination of the two non-parallel planes, generates all planes containing the line of intersection (the "hinge"). After students learn about subspaces of Rn, this can be justified by considering the linearly independent normal vectors n1 = < a1b1c1 > and n2 = < a2b2c2 >, which span a two-dimensional subspace of R3 - the plane perpendicular to the line of intersection. Therefore, any plane containing the line of intersection has a normal vector that can be expressed as a linear combination pn1 + qn2 for some values p and q.
  • In a row operation where k times the i-th row is added to the j-th row, the normal vector of the j-th plane is replaced with the linear combination of "old" normal vectors nj + kni. The set of all such linear combinations does not form the entire 2-dimensional subspace mentioned above, as the vectors collinear with ni are outside the set. Of course, this is the intended behavior of elementary row operations to prevent one of the equations from being overwritten with another.
  • One aspect of elementary row operations that remains completely transparent under this illustration is the scaling of the matrix rows (and the corresponding normals). Such scaling is one of the outcomes of row operations of the type
    rowj + (k)rowi → rowj
    (in addition to the resulting rotation). Scaling can also be explicitly executed by performing an operation
    (k)rowi → rowi
  • The third type of an elementary row operation,
    rowi ↔ rowj
    can, in the context of the HINGES puzzle, be viewed as resulting in altering the "door/wall/ghost plane" assignment sequence.

The HINGES activity could be used to compare geometrically the Gauss-Jordan reduction procedure (which transforms the augmented matrix to its reduced row echelon form) to Gaussian elimination (where row echelon form is used instead).

Also note that the rotation and scaling of a plane -- and of the corresponding normal vector -- can be discussed in a more concrete setting of linear transformations in R3 after this material has been reached -- see, e.g., (Anton, 2000, pp. 180-181).

Przemyslaw Bogacki, "HINGES - An Illustration of Gauss-Jordan Reduction - Outcomes and Solution," Convergence (June 2005)