Answer Key — Eigenvalues and orthogonality

Linear algebra

Do not share with students before practice tests.

Worksheets — Sets 1–10

| Set | Guidance | | --- | --- | | Set 1 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 2 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 3 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 4 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 5 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 6 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 7 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 8 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 9 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 10 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. |

Worksheets — Sets 11–20

| Set | Guidance | | --- | --- | | Set 11 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 12 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 13 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 14 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 15 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 16 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 17 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 18 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 19 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. | | Set 20 | Eigenvalues and orthogonality — verify written work and accept reasonable drawings. |

Review tiers 1–10

• Checklists are parent-judged.
• Written items: verify against practice set patterns.
• Standard text: Eigenvalues and eigenvectors; diagonalization of square matrices Orthogonality, projections, and the Gram–Schmidt process Least squares solutions to inconsistent systems Inner product spaces and orthonormal bases (introduction)

Practice test scoring

| Tier | Pass guidance | | --- | --- | | 1–3 | Most oral tasks smooth; majority of written correct | | 4–6 | 7/10+ total with clear understanding | | 7–8 | 9/12+ with explanations | | 9–10 | Near-perfect; ready to move on |

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