Linear Algebra I: systems, spaces and linear maps
Sixteen undergraduate foundation lessons connecting elimination, vector spaces, linear maps, determinants, eigenvectors, orthogonality and least squares. Written arguments require review. Linear Algebra II will extend abstract operator theory, spectral results, quadratic and canonical forms, and singular-value analysis.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
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Linear algebra: systems, maps and evidence assessment
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Calibration dossier: noisy differences around a network
Sustained investigation with separate transfer evidence, saved revisions and human review. It does not change lesson participation requirements.
Lesson sequence
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Solve systems without changing their solutions
Lesson details and objectives
Explain elementary row operations and trace elimination in an augmented matrix.
- Explain elementary row operations and trace elimination in an augmented matrix.
- Check a fresh case and justify the method's assumptions.
6 practice tasks.
Distinguish no solution, one solution and a family
Lesson details and objectives
Read pivots, contradictions and free variables without counting equations alone.
- Read pivots, contradictions and free variables without counting equations alone.
- Check a fresh case and justify the method's assumptions.
6 practice tasks.
Multiply matrices in the order the maps act
Lesson details and objectives
Track shapes, entries and noncommuting products.
- Track shapes, entries and noncommuting products.
- Check a fresh case and justify the method's assumptions.
6 practice tasks.
Undo a square linear system when possible
Lesson details and objectives
Construct inverses by elimination and recognize singularity.
- Construct inverses by elimination and recognize singularity.
- Check a fresh case and justify the method's assumptions.
6 practice tasks.
Build a space using linear combinations
Lesson details and objectives
Test subspace closure and describe spans, including polynomial examples.
- Test subspace closure and describe spans, including polynomial examples.
- Give a general argument as well as a numerical example.
6 practice tasks.
Detect redundancy using the zero combination
Lesson details and objectives
Separate a dependence relation from a representation of a nonzero target.
- Separate a dependence relation from a representation of a nonzero target.
- Give a general argument as well as a numerical example.
6 practice tasks.
Choose enough directions without redundancy
Lesson details and objectives
Construct bases for coordinate, solution and polynomial spaces.
- Construct bases for coordinate, solution and polynomial spaces.
- Give a general argument as well as a numerical example.
6 practice tasks.
Describe what a linear map loses and reaches
Lesson details and objectives
Connect abstract maps with matrices, kernels and images.
- Connect abstract maps with matrices, kernels and images.
- Give a general argument as well as a numerical example.
6 practice tasks.
Account for lost and retained dimensions
Lesson details and objectives
Prove rank-nullity and use pivots without confusing row and column spaces.
- Prove rank-nullity and use pivots without confusing row and column spaces.
- Connect the computation with a statement about linear maps.
6 practice tasks.
Change coordinates without changing the vector
Lesson details and objectives
Use basis matrices and distinguish a vector from its coordinate list.
- Use basis matrices and distinguish a vector from its coordinate list.
- Connect the computation with a statement about linear maps.
6 practice tasks.
Use determinants to detect collapse and track signed volume
Lesson details and objectives
Compute determinants with row-operation rules and qualify geometric interpretations.
- Compute determinants with row-operation rules and qualify geometric interpretations.
- Connect the computation with a statement about linear maps.
6 practice tasks.
Find directions preserved by a linear map
Lesson details and objectives
Compute eigenspaces and distinguish zero eigenvalues from forbidden zero eigenvectors.
- Compute eigenspaces and distinguish zero eigenvalues from forbidden zero eigenvectors.
- Connect the computation with a statement about linear maps.
6 practice tasks.
Compute in an eigenvector basis when one exists
Lesson details and objectives
Distinguish diagonalizability from merely having characteristic roots.
- Distinguish diagonalizability from merely having characteristic roots.
- Verify the result and state its existence or uniqueness assumptions.
6 practice tasks.
Split a vector into parallel and perpendicular parts
Lesson details and objectives
Derive orthogonal projection and explain why it gives the nearest point.
- Derive orthogonal projection and explain why it gives the nearest point.
- Verify the result and state its existence or uniqueness assumptions.
6 practice tasks.
Remove old directions before adding a new one
Lesson details and objectives
Construct an orthogonal basis, detect dependence and connect the result to QR.
- Construct an orthogonal basis, detect dependence and connect the result to QR.
- Verify the result and state its existence or uniqueness assumptions.
6 practice tasks.
Fit an inconsistent system without hiding the error
Lesson details and objectives
Derive normal equations, examine uniqueness and distinguish fit from explanation.
- Derive normal equations, examine uniqueness and distinguish fit from explanation.
- Verify the result and state its existence or uniqueness assumptions.
6 practice tasks.