Operators: structure, geometry and justified decompositions
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 19 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.
Choose a method and write the argument before opening guidance. State the field, domain, dimensions and theorem hypotheses. A computed example does not prove a general assertion.
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For aₙ=3+(n−1)2ⁿ, derive a homogeneous constant-coefficient recurrence and prove its minimum possible order for all n≥0. Give enough initial values to specify it. Then remove the n2ⁿ term: how does the minimum order change, and what cannot be concluded from a matching finite prefix alone?
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.
At nodes0,1,3,4 the reported values are0,1,9,17. Construct an exact functional proving no quadratic fits them, verify it on every quadratic monomial, and repair only the last observation while retaining the anchors. Is a cubic fit possible, and does it identify a faulty observation? Finally determine the least degree for alternating zero/one values on seven increasing real nodes.
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.
On R⁴ prescribe Te₁=(1,2,3,0) and Te₂=(2,−1,0,4). Construct every self-adjoint extension and count its real parameters. Can the orthogonal complement of span(e₁,e₂) have an eigenbasis made of its own vectors? Add the observation T(e₁+e₂)=(3,1,3,5): can any free parameter repair it?
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:C²→C have row (1,i), input Gram matrix diag(1,4) and output Gram scalar3. Compute its adjoint and verify the identity for x=(1,1), y=1. Change input coordinates by K=diag(i,1) and output coordinates by L=2. Compute A′, both new Gram matrices and B′. Compare with ordinary transpose and explain why the physical map is unchanged.
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.
On [−2,2], use Lp=((x²−4)p′)′ to construct an orthogonal basis for P₂. Find the best integral-squared-error approximation to |x| in P₂ and its error. Would adding cubic terms improve it? Is the value at zero exact? State which argument establishes optimality.
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.
Count all 3 × 3 matrices A with A²=I over F₃ and over F₂. Give a complete structural argument; do not extrapolate the two-dimensional counts. Explain which roots are diagonalizable.
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:R²→R³ have rows (2,2), (1,1), (2,2). Construct its positive polar factor and canonical partial isometry, compute the norm and every maximizing unit input, and give two distinct isometric extensions. Then transpose A: can its canonical factor extend to an isometry on all of R³?
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.
Over F₅ find every polynomial of degree at most four taking values 1,4,1 at inputs 1,2,4. Count the family. Add the observations p(0)=3 and then p(3)=4, tracking the remaining freedom after each. Finally raise the degree cap to five while retaining all five field values: is the formal polynomial still unique?
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 C³ let A have columns (1,i,0) and (1,0,i). Construct its normalized thin QR and U=R⁻¹. Check both conjugate orthogonality and reconstruction. Apply D=diag(i,−1) to Q and compensate in R: is the resulting factorization normalized? Finally replace the second input column by twice the first and explain the changed existence/uniqueness conclusion.
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 P₄ be the real polynomials of degree at most four and T:P₄→P₂ the second-derivative map. Find kernel and image, construct a direct complement to the kernel, and prove that the induced map P₄/ker T→P₂ is an isomorphism. Determine the annihilator of W={p:p(0)=p′(0)=0} in P₄*.
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.
For A with rows (2,1,0),(0,2,0),(0,0,3), find invariant coordinate subspaces, characteristic and minimal polynomials, and eigenspace dimensions. Prove why the repeated root in the minimal polynomial cannot be removed.
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.
For the preceding matrix A, derive A⁻¹ as a polynomial of degree at most two. Construct a polynomial projection onto the 3-eigenspace along the generalized 2-eigenspace and verify idempotence and its image.
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.
A nilpotent operator N on a seven-dimensional complex space has kernel dimensions 3,5,6,7 for N,N²,N³,N⁴. Determine its Jordan block sizes and minimal polynomial. Explain why these dimensions determine the sizes but not a unique Jordan basis.
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.
Use the convention ⟨x,y⟩=Σxⱼ conjugate(yⱼ). Find the Riesz vector for f(z)=2z₁−iz₂. Project x=(1,1) onto the span of u=(1,i), verify orthogonality, and explain the conjugate-linearity of the map from Riesz vectors to functionals.
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.
For A:C³→C² with rows (1,i,0),(0,0,2), compute A*, ker A and im A*. Prove generally that im A*=(ker A)⊥ in finite dimensions. Then explain why a normal upper-triangular complex matrix must be diagonal.
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.
For B with rows (6,2),(2,6), derive its orthonormal eigenbasis, exact Rayleigh bounds and positive square root. Explain which part of the argument guarantees uniqueness of the positive square root.
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 C=diag(1,1,−1) and D have rows (2,i,0),(−i,2,0),(0,0,5). Construct a common orthonormal eigenbasis and explain the general criterion for two complex normal operators to admit one.
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.
For A with rows (0,0),(4,0),(0,1), find its singular values, a full SVD and pseudoinverse. Solve least squares for b=(2,8,3), identify the residual, and give a rank-one approximation with its operator-norm error.
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=diag(5,0.02), b=(5,0), and perturb b by δb=(0,0.01). Compute the 2-norm condition number and both relative changes. Explain why a small residual is not sufficient evidence of small forward error, and compare normal equations with QR/SVD for least squares.
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
For aₙ=3+(n−1)2ⁿ, derive a homogeneous constant-coefficient recurrence and prove its minimum possible order for all n≥0. Give enough initial values to specify it. Then remove the n2ⁿ term: how does the minimum order change, and what cannot be concluded from a matching finite prefix alone?
At nodes0,1,3,4 the reported values are0,1,9,17. Construct an exact functional proving no quadratic fits them, verify it on every quadratic monomial, and repair only the last observation while retaining the anchors. Is a cubic fit possible, and does it identify a faulty observation? Finally determine the least degree for alternating zero/one values on seven increasing real nodes.
On R⁴ prescribe Te₁=(1,2,3,0) and Te₂=(2,−1,0,4). Construct every self-adjoint extension and count its real parameters. Can the orthogonal complement of span(e₁,e₂) have an eigenbasis made of its own vectors? Add the observation T(e₁+e₂)=(3,1,3,5): can any free parameter repair it?
Let A:C²→C have row (1,i), input Gram matrix diag(1,4) and output Gram scalar3. Compute its adjoint and verify the identity for x=(1,1), y=1. Change input coordinates by K=diag(i,1) and output coordinates by L=2. Compute A′, both new Gram matrices and B′. Compare with ordinary transpose and explain why the physical map is unchanged.
On [−2,2], use Lp=((x²−4)p′)′ to construct an orthogonal basis for P₂. Find the best integral-squared-error approximation to |x| in P₂ and its error. Would adding cubic terms improve it? Is the value at zero exact? State which argument establishes optimality.
Count all 3 × 3 matrices A with A²=I over F₃ and over F₂. Give a complete structural argument; do not extrapolate the two-dimensional counts. Explain which roots are diagonalizable.
Let A:R²→R³ have rows (2,2), (1,1), (2,2). Construct its positive polar factor and canonical partial isometry, compute the norm and every maximizing unit input, and give two distinct isometric extensions. Then transpose A: can its canonical factor extend to an isometry on all of R³?
Over F₅ find every polynomial of degree at most four taking values 1,4,1 at inputs 1,2,4. Count the family. Add the observations p(0)=3 and then p(3)=4, tracking the remaining freedom after each. Finally raise the degree cap to five while retaining all five field values: is the formal polynomial still unique?
In C³ let A have columns (1,i,0) and (1,0,i). Construct its normalized thin QR and U=R⁻¹. Check both conjugate orthogonality and reconstruction. Apply D=diag(i,−1) to Q and compensate in R: is the resulting factorization normalized? Finally replace the second input column by twice the first and explain the changed existence/uniqueness conclusion.
Let P₄ be the real polynomials of degree at most four and T:P₄→P₂ the second-derivative map. Find kernel and image, construct a direct complement to the kernel, and prove that the induced map P₄/ker T→P₂ is an isomorphism. Determine the annihilator of W={p:p(0)=p′(0)=0} in P₄*.
For A with rows (2,1,0),(0,2,0),(0,0,3), find invariant coordinate subspaces, characteristic and minimal polynomials, and eigenspace dimensions. Prove why the repeated root in the minimal polynomial cannot be removed.
For the preceding matrix A, derive A⁻¹ as a polynomial of degree at most two. Construct a polynomial projection onto the 3-eigenspace along the generalized 2-eigenspace and verify idempotence and its image.
A nilpotent operator N on a seven-dimensional complex space has kernel dimensions 3,5,6,7 for N,N²,N³,N⁴. Determine its Jordan block sizes and minimal polynomial. Explain why these dimensions determine the sizes but not a unique Jordan basis.
Use the convention ⟨x,y⟩=Σxⱼ conjugate(yⱼ). Find the Riesz vector for f(z)=2z₁−iz₂. Project x=(1,1) onto the span of u=(1,i), verify orthogonality, and explain the conjugate-linearity of the map from Riesz vectors to functionals.
For A:C³→C² with rows (1,i,0),(0,0,2), compute A*, ker A and im A*. Prove generally that im A*=(ker A)⊥ in finite dimensions. Then explain why a normal upper-triangular complex matrix must be diagonal.
For B with rows (6,2),(2,6), derive its orthonormal eigenbasis, exact Rayleigh bounds and positive square root. Explain which part of the argument guarantees uniqueness of the positive square root.
Let C=diag(1,1,−1) and D have rows (2,i,0),(−i,2,0),(0,0,5). Construct a common orthonormal eigenbasis and explain the general criterion for two complex normal operators to admit one.
For A with rows (0,0),(4,0),(0,1), find its singular values, a full SVD and pseudoinverse. Solve least squares for b=(2,8,3), identify the residual, and give a rank-one approximation with its operator-norm error.
Let A=diag(5,0.02), b=(5,0), and perturb b by δb=(0,0.01). Compute the 2-norm condition number and both relative changes. Explain why a small residual is not sufficient evidence of small forward error, and compare normal equations with QR/SVD for least squares.
