Linear algebra: systems, maps and evidence assessment
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 130 minutes, as an unvalidated planning estimate. This collection contains 10 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 methods without opening the worked solutions first. Use paper for diagrams and calculations; write enough reasoning to justify assumptions and every requested conclusion.
The suggested working time is guidance, not a timer. Written and symbolic responses require human review; a recorded response is not a proof-quality grade.
This public mixed practice samples the published unit. It is not a secure examination, complete AP paper or guarantee of course readiness. Saved first responses and known answer exposure persist across retries and imports.
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0 of 10 tasks have recorded responses. No overall mastery score is assigned.
Classify the solutions of x+y+z=3, x+2y+3z=4, 2x+3y+4z=k as k varies. Give a parameterization when consistent and explain the augmented-row contradiction otherwise.
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=[[1,2],[0,1]] and B=[[2,0],[1,1]], acting on column vectors. Compute AB and BA, decide whether order matters, and calculate (AB)⁻¹. Explain the order of inverse operations.
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 W={(x,y,z)∈R³:x−2y+z=0}. Prove W is a subspace, construct a basis and determine its dimension. Is the parallel set x−2y+z=1 a 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.
For T:R³→R² defined by T(x,y,z)=(x+y,y+z), find bases for kernel and image, the rank and nullity, and decide injectivity and surjectivity. Verify rank-nullity using the domain dimension.
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 R² let basis B=((1,1),(1,−1)) and basis C=((2,0),(1,1)). A vector has B-coordinates(3,−2). Find its ordinary coordinates and its C-coordinates. Derive the matrix that converts B-coordinates to C-coordinates.
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 linear transformation in R³ has matrix [[2,1,0],[0,−3,1],[0,0,4]]. Find its determinant, explain its volume and orientation effects, and decide invertibility. Does a negative determinant mean a geometric volume is negative?
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=[[3,1],[0,2]], find eigenvalues and a basis of eigenvectors. Construct A=PDP⁻¹ and use it to derive a formula for Aⁿ for every nonnegative integer n, checking n=0 and n=1.
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=(3,1,2) and u=(1,1,0), compute the orthogonal projection p of b onto span{u} and residual r=b−p. Prove p minimizes distance from b to that line using a Pythagorean decomposition.
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.
Apply Gram–Schmidt to a₁=(1,0,1), a₂=(1,2,1), in that order. Give an orthonormal pair, a thin QR factorization of the 3×2 matrix with these columns, and explain what would fail if the second column were2a₁.
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.
Fit y≈a+bx by least squares to (0,2),(1,1),(2,3). Derive the normal equations, solve for a,b, and verify the residual is perpendicular to both design columns. What does the residual tell you about exact interpolation?
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
Classify the solutions of x+y+z=3, x+2y+3z=4, 2x+3y+4z=k as k varies. Give a parameterization when consistent and explain the augmented-row contradiction otherwise.
Let A=[[1,2],[0,1]] and B=[[2,0],[1,1]], acting on column vectors. Compute AB and BA, decide whether order matters, and calculate (AB)⁻¹. Explain the order of inverse operations.
For T:R³→R² defined by T(x,y,z)=(x+y,y+z), find bases for kernel and image, the rank and nullity, and decide injectivity and surjectivity. Verify rank-nullity using the domain dimension.
In R² let basis B=((1,1),(1,−1)) and basis C=((2,0),(1,1)). A vector has B-coordinates(3,−2). Find its ordinary coordinates and its C-coordinates. Derive the matrix that converts B-coordinates to C-coordinates.
A linear transformation in R³ has matrix [[2,1,0],[0,−3,1],[0,0,4]]. Find its determinant, explain its volume and orientation effects, and decide invertibility. Does a negative determinant mean a geometric volume is negative?
For A=[[3,1],[0,2]], find eigenvalues and a basis of eigenvectors. Construct A=PDP⁻¹ and use it to derive a formula for Aⁿ for every nonnegative integer n, checking n=0 and n=1.
For b=(3,1,2) and u=(1,1,0), compute the orthogonal projection p of b onto span{u} and residual r=b−p. Prove p minimizes distance from b to that line using a Pythagorean decomposition.
Apply Gram–Schmidt to a₁=(1,0,1), a₂=(1,2,1), in that order. Give an orthonormal pair, a thin QR factorization of the 3×2 matrix with these columns, and explain what would fail if the second column were2a₁.
Fit y≈a+bx by least squares to (0,2),(1,1),(2,3). Derive the normal equations, solve for a,b, and verify the residual is perpendicular to both design columns. What does the residual tell you about exact interpolation?
