Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 150 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt these three original public proof problems before reading the worked solutions. Show field arithmetic, counting arguments and general justifications.
Keep ordered bases, subspaces and affine fibers distinct. No secure examination or independently verified mastery is implied by saving a response.
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0 of 3 tasks have recorded responses. No overall mastery score is assigned.
Over F3 let T:F3⁴→F3² have rows (1,1,0,1) and (0,1,1,2). Prove surjectivity, parametrize T⁻¹(1,2), and identify the kernel with an explicit basis. Count the elements of every fiber, the number of fibers and the ordered bases of the kernel. Explain why a basis count cannot be used as the number of solutions of the inhomogeneous equation, and why the fiber over (1,2) is not itself a vector subspace.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In V=F2⁴ fix the line L=span{(1,0,0,0)}. Count all two-dimensional subspaces W, those containing L, and those satisfying W∩L={0}. Give proofs by counting ordered independent pairs and by passing to V/L, justifying why the quotient correspondence is a bijection. If π:V→V/L is the quotient map, determine the rank of π restricted to each kind of W. Finally explain why counting arbitrary pairs of nonzero vectors without excluding dependence gives a wrong answer.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let A have integer rows (0,2),(1,0). Compare its reductions over F3, F7 and F2: compute eigenvalues, eigenspaces and diagonalizability. Count the invariant lines over F7. Over F3 construct the quadratic extension with j²=2 and explicitly diagonalize A there, proving that the proposed quotient is a field. Diagnose the claim that an extension of scalars and a change of prime modulus are the same operation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Over F3 let T:F3⁴→F3² have rows (1,1,0,1) and (0,1,1,2). Prove surjectivity, parametrize T⁻¹(1,2), and identify the kernel with an explicit basis. Count the elements of every fiber, the number of fibers and the ordered bases of the kernel. Explain why a basis count cannot be used as the number of solutions of the inhomogeneous equation, and why the fiber over (1,2) is not itself a vector subspace.
In V=F2⁴ fix the line L=span{(1,0,0,0)}. Count all two-dimensional subspaces W, those containing L, and those satisfying W∩L={0}. Give proofs by counting ordered independent pairs and by passing to V/L, justifying why the quotient correspondence is a bijection. If π:V→V/L is the quotient map, determine the rank of π restricted to each kind of W. Finally explain why counting arbitrary pairs of nonzero vectors without excluding dependence gives a wrong answer.
Let A have integer rows (0,2),(1,0). Compare its reductions over F3, F7 and F2: compute eigenvalues, eigenspaces and diagonalizability. Count the invariant lines over F7. Over F3 construct the quadratic extension with j²=2 and explicitly diagonalize A there, proving that the proposed quotient is a field. Diagnose the claim that an extension of scalars and a change of prime modulus are the same operation.
