Learn Solve modular systems, count bases and subspaces, and test spectral claims with the scalar field stated explicitly. These finite-field models retain the distinction between linear algebra and arithmetic over composite residue rings.
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Elimination must use legal inverses rather than ordinary fraction division.
Modulo a prime p every nonzero residue has a multiplicative inverse. Modulo a composite integer a nonzero element may not be invertible; use the gcd criterion.
Finite-field counting turns a basis of free directions into an exact number of choices.
For a linear map with n input coordinates and rank r, the kernel has dimension n−r. A nonempty solution set is a translate of that kernel, not usually a subspace itself.
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Systems, subspaces and changing scalars
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Use reversible row operations to classify modular systems, then prove how rank counts both attainable outputs and solutions above each output.
6 practice tasks.
Count independent vectors sequentially, then remove the exact multiplicity of bases to count subspaces over a finite field.
6 practice tasks.
Keep the scalar field visible while solving eigenvector equations, constructing extension fields and testing diagonalization.
6 practice tasks.