# LA Endterm Study Plan (Recency-Weighted)

## Tier 1: 4+ recent appearances (2023-2024) — dependency order

| # | Topic | 2024 Endterm Q | Pattern | Lectures | Done |
|---|-------|---------------|---------|----------|------|
| 1 | Linear systems (with parameter) | Q1 | P1 | L01, L03 | [x] |
| 2 | Eigenvalues & eigenvectors | Q4a, Q7a/b | P10 | L11 | [x] |
| 3 | Matrix equation solving | Q11 | P2 | L05, L06 | [x] |
| 4 | Linear transformations | — | P5 | L04 | [ ] |
| 5 | Coordinate vectors | — | P8 | L09 | [ ] |
| 6 | Orthogonal projection | Q5b | P14 | L16 | [x] |
| 7 | Least-squares / best fit line | Q8 | P13 | L19 | [x] |
| 8 | Discrete dynamical systems | Q2 | P12 | L14 | [x] |
| 9 | Academic reasoning | Q12, Q13 | P24 | L07, L17 | [ ] |

## Tier 2: 2-3 recent appearances

| # | Topic | 2024 Endterm Q | Pattern | Lectures | Done |
|---|-------|---------------|---------|----------|------|
| 10 | Orthogonal complement | Q9 | P16 | L15 | [x] |
| 11 | Determinant properties | Q11 | P18 | L10 | [x] |
| 12 | Invertibility/det from equations | Q11 | P21 | L06, L10 | [x] |
| 13 | Range/image of transformation | — | P22 | L04, L08 | [ ] |
| 14 | Diagonalizability (with parameter) | Q4b | P11 | L12 | [x] |
| 15 | Gram-Schmidt | Q5a | P15 | L18 | [x] |
| 16 | Complex eigenvalues / PCP^-1 | Q10 | P20 | L13 | [x] |
| 17 | Column space basis | Q3 | P7 | L08, L09 | [x] |
| 18 | Determinant computation | — | P3 | L10 | [x] |
| 19 | Matrix inverse | — | P4 | L06 | [ ] |
| 20 | Null space / rank-nullity | — | P6 | L08, L09 | [x] |
| 21 | Linear independence | — | P9 | L03 | [ ] |

## Tier 3: Skip unless you have time

| # | Topic | 2024 Endterm Q | Pattern | Lectures | Done |
|---|-------|---------------|---------|----------|------|
| 22 | Matrix powers via diag. | — | P19 | L12 | [ ] |
| 23 | Subspace identification | — | P17 | L08 | [ ] |
| 24 | LU decomposition | — | P23 | — | [ ] |
| 25 | Orthonormal matrix props | — | P25 | L20 | [ ] |

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## Endterm 2025 Questions

| Q# | Pts | Topic | Pattern | Lectures | Done |
|----|-----|-------|---------|----------|------|
| 1a | 3 | RREF of a matrix | P1 | L01 | [x] |
| 1b | 3 | rank(A) and dim Nul(A) | P6 | L08, L09 | [x] |
| 2 | 3 | Standard matrix of linear transformation (given T on non-standard vectors) | P5 | L04 | [x] |
| 3 | 3 | Singular matrix — find parameter relation (multiple choice) | P21 | L10 | [x] |
| 4 | 3 | Express A^-1 from polynomial identity A^T A + 2A = 3I | P2 | L05, L06 | [x] |
| 5 | 3 | Which P works for diagonalization A = PDP^-1 (multiple choice) | P11 | L12 | [x] |
| 6a | 3 | Complex eigenvalues — rotation-scaling form (r and phi) | P20 | L13 | [x] |
| 6b | 3 | Eigenvector for complex eigenvalue | P10, P20 | L11, L13 | [x] |
| 7a | 3 | General solution of discrete dynamical system (given PDP^-1) | P12 | L14 | [x] |
| 7b | 3 | IVP solution for dynamical system | P12 | L14 | [x] |
| 8 | 3 | Second row of projection matrix (proj onto W) | P14 | L16 | [x] |
| 9 | 3 | Orthogonal basis for Span{b1, b2, b3} (Gram-Schmidt) | P15 | L18 | [x] |
| 10a | 3 | Normal equations for least-squares line | P13 | L19 | [x] |
| 10b | 3 | Best fit line | P13 | L19 | [x] |
| 11 | 3 | Eigenspace basis from symmetric matrix + orthogonality | P10, P25 | L11, L20 | [x] |
| 12 | 3 | Academic reasoning: Ax=0 trivial => 0 not eigenvalue of A^T | P24 | L07, L17 | [ ] |
| 13 | 3 | Academic reasoning: AB=0 => BA=0? | P24 | L07, L17 | [ ] |

---

## How to Recognize Each Pattern (exact question phrasings)

### P1 — Linear Systems
- "Find the common intersection point, if it exists"
- "For which value(s) of h is the system consistent/inconsistent"
- "Find the reduced echelon form of the above matrix"
- "How many free variables does the system have"

### P2 — Matrix Equation Solving
- "Express A^-1 in terms of A, A^T and I"
- "Solve for X" (given equation with inverses/transposes)
- "Suppose A satisfies the identity ... Express A^-1"
- Trick: multiply both sides by A^-1 on the correct side

### P3 — Determinant Computation
- "Compute det(A)" / "Find the determinant"
- "If det[...] = x, find det(-3B)" (with modified matrix)
- Expand along row/column with most zeros

### P5 — Linear Transformations
- "Find the standard matrix A of this transformation"
- "Write NSI if there is not sufficient information"
- Given T on non-standard vectors: express e1, e2 as combinations, then use linearity
- Rotation/reflection/shear compositions: multiply matrices right-to-left

### P6 — Null Space / Rank-Nullity
- "Find rank(A) and dim Nul(A)"
- "Find dim(Nul A)" given matrix dimensions and rank
- rank + nullity = number of columns

### P7 — Column Space Basis
- "Give all those that are guaranteed to be a basis for the column space"
- "Find a basis for Col(A)"
- Pivot columns of RREF tell you WHICH columns, take them from ORIGINAL A

### P8 — Coordinate Vectors
- "Find [w]_B" / "Find the coordinate vector"
- "Does w lie in Span(B)?"
- Row reduce [b1 b2 ... | w]

### P9 — Linear Independence
- "For which value(s) of alpha is the set linearly dependent"
- "Find dim(Span{...})"
- det = 0 means dependent (square case)

### P10 — Eigenvalues & Eigenvectors
- "Find all the eigenvalues"
- "Find a basis for the eigenspace"
- "Find an eigenvector associated to lambda = ..."
- Solve det(A - λI) = 0, then (A - λI)x = 0

### P11 — Diagonalizability
- "For which value(s) of h is A diagonalizable?"
- "Which of the following matrices P can be used in a correct diagonalization?"
- Check: g.m. = a.m. for each eigenvalue
- Columns of P must be eigenvectors of A

### P12 — Discrete Dynamical Systems
- "Find the general solution of x_{k+1} = Ax_k"
- "Find the unique solution of the initial value problem"
- "Find y_n"
- x_k = c1 λ1^k v1 + c2 λ2^k v2, use x_0 to find c1, c2

### P13 — Least-Squares / Best Fit Line
- "Find the equation y = a + bx of the best line (in the least-squares sense)"
- "Give the normal equations"
- Build design matrix X (column of 1s, column of x-values), solve X^T X β = X^T y

### P14 — Orthogonal Projection
- "Find the second column/row of the standard matrix P of the orthogonal projection"
- "Let T(y) = proj_W(y). Find the standard matrix"
- "Find proj_W(y)"
- Gram-Schmidt first if needed, then P = UU^T (orthonormal columns)

### P15 — Gram-Schmidt
- "Find an orthogonal basis for Col(A)" / "for Span{b1, b2, b3}"
- "Find an orthogonal basis for the column space"
- v_k = b_k minus projections onto all previous v's

### P16 — Orthogonal Complement
- "Find a basis for W^perp"
- "Find a vector orthogonal to Col(A)"
- Put spanning vectors as ROWS, find null space

### P20 — Complex Eigenvalues / PCP^-1
- "It is given that λ = a+bi is an eigenvalue. Find P and C such that A = PCP^-1"
- "A is similar to a matrix of the form r[cos φ, -sin φ; sin φ, cos φ]. Find r and φ"
- r = sqrt(a^2+b^2), φ = arctan(b/a)
- P = [Re(v) | Im(v)] from eigenvector for complex eigenvalue

### P21 — Invertibility / det from Equations
- "Find all possible values for det(A)" given a matrix equation like A^3 = -5(A^T)^-1
- "For which values of a and b is the matrix singular?"
- Take det of both sides, use det(AB) = det(A)det(B), det(A^T) = det(A)

### P24 — Academic Reasoning
- "If you think the statement is true, give a proof"
- "If you think the statement is false, give an explicit counterexample"
- Common true: use P^2=P, expand dot products, express A^-1 from equation
- Common false: try simple 2x2 matrices with 0s and 1s

### P25 — Symmetric Matrix / Orthogonal Diagonalization
- "Find an orthogonal basis of R^n consisting of eigenvectors"
- "It is given that A is symmetric... find a basis for the eigenspace"
- Symmetric → eigenspaces for different eigenvalues are orthogonal
- If eigenspace E_λ is known, E_μ is its orthogonal complement

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## Your Confidence Assessment (based on this session)

### Strong — solved with minimal hints
- Least-squares / best fit line (Q8) — nailed the process
- Discrete dynamical systems (Q2, Q7) — smooth after initial explanation
- Determinant properties / det from equations (Q11) — one-step
- Gram-Schmidt (Q5a, Q9) — mechanical, no issues
- Column space basis (Q3) — understood after correction
- Diagonalizability (Q4b, Q5 2025) — solid
- Orthogonal complement (Q9 2024, Q11 2025) — rows → null space, got it
- Complex eigenvalues (Q10, Q6 2025) — reliable once you learned the process
- Express A^-1 from identity (Q4 2025) — instant

### Needs review — required multiple hints or conceptual confusion
- Orthogonal projection / projection matrix (Q5b) — confused about P = UU^T, what y means, what "second column" means. RE-READ L16 summary tomorrow
- Eigenvectors from scratch — needed reminders on free variables, complex arithmetic. Practice one more from scratch tomorrow
- Linear transformations (Q2 2025) — needed hint about expressing e1, e2 as combinations. Review L04 standard matrix method

### Not attempted — review if time permits
- Academic reasoning (Q12, Q13 on both exams)
- Coordinate vectors (P8)
- Orthogonal diag. of symmetric matrix (Q6 2024)
- Linear transformations with rotation/reflection (P5 geometric)