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Use randomized questions for active recall, flashcards for vocabulary, generated numerical problems for fluency, and the checklist for a final self-audit. Progress in this page is local to the browser and does not submit grades.
Randomized cumulative quiz
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Assessment generator
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Concept flashcards
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Core vocabulary deck
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Generated calculation practice
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Numerical problem generator
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Searchable glossary
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| Term | Definition | Connection |
|---|---|---|
| Basis | A linearly independent spanning set. | Defines unique coordinates; its size is dimension. |
| Column space | The span of a matrix’s columns. | Equals the image of x↦Ax. |
| Condition number | A measure of relative sensitivity to perturbations. | For the 2-norm, κ=σmax/σmin. |
| Determinant | Signed volume-scaling factor of a square map. | Nonzero exactly when the matrix is invertible. |
| Diagonalization | A=PDP⁻¹ with D diagonal. | Possible when A has an eigenvector basis. |
| Eigenvalue | Scalar λ satisfying Av=λv for some v≠0. | Controls scaling along an invariant direction. |
| Gram–Schmidt | Procedure producing an orthonormal basis from independent vectors. | Leads to QR factorization. |
| Image | Set of all outputs of a transformation. | For a matrix map, the image is C(A). |
| Kernel | All inputs mapped to zero. | Its dimension is nullity. |
| Least squares | Minimization of ‖Ax−b‖. | Residual is orthogonal to C(A). |
| Linear independence | Only the trivial coefficient combination yields zero. | Eliminates redundant generating directions. |
| Linear transformation | A map preserving addition and scalar multiplication. | Represented by a matrix after bases are chosen. |
| Nullity | Dimension of the null space. | rank+nullity=number of columns. |
| Orthogonal | Having zero inner product. | Supports independent decomposition and projection. |
| Projection | Closest component of a vector in a subspace. | Projection matrices are symmetric and idempotent for orthogonal projection. |
| Rank | Dimension of the column space. | Equals pivot count and row-space dimension. |
| RREF | Canonical row-reduced form. | Exposes pivots, free variables, and consistency. |
| Singular value | Nonnegative principal stretch magnitude. | Square root of an eigenvalue of AᵀA. |
| Span | Set of all linear combinations. | Smallest subspace containing the generators. |
| Trace | Sum of diagonal entries. | Equals sum of eigenvalues with multiplicity. |
Exam-preparation checklist
Do not mark the page complete until these tasks are comfortable
Computational fluency
- Multiply compatible matrices and explain every output entry.
- Row-reduce and write complete parametric solutions.
- Compute small determinants and inverses.
- Find bases for span, column space, and null space.
- Compute projections and perform Gram–Schmidt.
- Find 2×2 eigenpairs and test diagonalizability.
- Set up a least-squares design matrix.
Conceptual fluency
- Explain why multiplication represents composition.
- State equivalent conditions for invertibility.
- Distinguish span, independence, basis, and dimension.
- Interpret kernel and image geometrically.
- Explain residual orthogonality.
- Describe eigenvectors as invariant directions.
- Interpret SVD as rotation–scaling–rotation.
Strong evidence of understanding
You can solve a problem, explain why the method applies, predict the form of the answer, and verify the result using a second representation or invariant.