Linear Algebra Students' Understanding of Similar Matrices and Matrix Representations of Linear Transformations in a MATLAB-Assisted Learning Environment

Abstract

The purpose of this research study was to understand how mathematics majors make sense of similar matrices, a linear algebra topic that is rich in internal and external connections, in a series of in-depth qualitative interviews in a technology-assisted (MATLAB) learning environment. The study was also an attempt to develop a cognitive theory in mathematics majors’ treatment of similar matrices and matrix representations of linear transformations in a computer-assisted learning environment by drawing primarily on three aspects of the experimental mathematics methodology, made possible by using modern digital tools to enhance learning: testing conjectures, suggesting proofs, and, computing lengthy hand derivations (Borwein & Bailey, 2003). The analysis revealed a diversity of innovative ways by which research participants coordinated analytic and computational (MATLAB) methods for understanding matrix similarity and its connections to the rest of linear algebra. The paper concludes by offering pedagogical implications and recommendations for further research.

Keywords:

• Matrix similarity
• trace and determinant
• eigentheory and diagonalization
• change of basis and transition matrices
• standard matrix of a linear transformation
• matrix representation of a linear transformation relative to a nonstandard basis
• MATLAB

Notes

1 This activity also included other versions with $T\left((x,y,z)\right)=(2x-2y+3z,3y-2z,-y+2z)$ and $T\left((x,y,z)\right)=(x+3y,3x+y,-2z)$.

2 (a) Obtain the transition matrix P from β′ to β via the $\left[\beta |\beta \prime \right]\stackrel{\text{rref}}{\to }\left[I|P\right]$ procedure; (b) Obtain the transition matrix P⁻¹ from β to β′ via the $\left[\beta \prime |\beta \right]\stackrel{\text{rref}}{\to }\left[I|P^{-1}\right]$ procedure; (3) Evaluate $A^{\prime }=P^{-1}AP$.

3 In this problem the eigenvalue -2 is of algebraic multiplicity 2.