CSE1205 Linear Algebra - Exam Patterns

1. Solving / Analyzing Linear Systems (with parameter) Very High

MID 2020 Q7 • 2023 Q1 • 2024 Q1 | END 2019 Q1 | RES 2019 Q1 • 2022 Q2 • 2023 Q1 • 2024 Q1

Given a linear system (often as an augmented matrix) with a parameter h or α, determine for which values the system is consistent, inconsistent, has a unique solution, infinitely many solutions, or a specific number of free variables.

Methods to Solve

Row reduce the augmented matrix [A | b] to echelon form, keeping the parameter symbolic
Look for contradictory rows (e.g., [0 0 ... 0 | k] with k ≠ 0) to find inconsistency conditions
Count pivot positions vs. free variables: #free vars = #columns - #pivots
For "exactly 1 free variable": need exactly (n-1) pivots and no contradiction in the augmented column
A system is consistent with infinitely many solutions iff it has at least one free variable and no contradictions

Example (Midterm 2024 Q1): For which value(s) of h is x₁a₁ + x₂a₂ + x₃a₃ = b inconsistent? Row reduce and find h = -5 or h = -3.

2. Matrix Equation Solving (Solve for X) Very High

MID 2020 Q4 • 2023 Q5 • 2024 Q5 | END 2019 Q16 • 2022 Q2 • 2023 Q4 • 2024 Q11 | RES 2022 Q3 • 2023 Q2 • 2024 Q4

Given an equation involving invertible matrices A, B, C, X with inverses and transposes, isolate X. Typically multiple-choice with tricky orderings.

Methods to Solve

Key rules: (AB)^-1 = B^-1A^-1, (A^T)^-1 = (A^-1)^T, (AB)^T = B^TA^T
Isolate X by multiplying both sides on the correct side (left or right) by the appropriate inverse
When transposes are involved: take the transpose of both sides if needed, remembering (AB)^T = B^TA^T
Order matters! Matrix multiplication is not commutative -- always multiply on the same side
Simplify fully: no expressions like (AB)^-1 in the final answer

Example (Midterm 2020 Q4): A^TB^-1(C^T+X)^T = C. Answer: X = (B^TA^-1 - I)C^T.

3. Determinant Computation Very High

MID 2020 Q3 • 2023 Q10 • 2024 Q10 | END 2019 Q14-15 | RES 2019 Q6 • 2022 Q7

Compute the determinant of a 3×3, 4×4, or 5×5 matrix. Often combined with det properties (column swaps, scalar multiplication).

Methods to Solve

Cofactor expansion: expand along a row/column with the most zeros
Row reduction to upper triangular, then det = product of diagonal entries (track sign changes from row swaps)
For triangular/block-triangular matrices: det = product of diagonal entries
Properties: det(cA) = cⁿdet(A) for n×n matrix; det(AB) = det(A)det(B); row swap flips sign; adding a multiple of one row to another doesn't change det
Column swap: changes sign of determinant

Example (Midterm 2024 Q6): If det[a b c; d e f; g h i] = x, find det(-3B) where B has columns permuted. Answer: (-3)³ · det(B) = -27x.

4. Matrix Inverse Computation High

MID 2020 Q5 • 2023 Q6 | END 2019 Q12 • 2022 Q3 • 2023 Q4 | RES 2022 Q5

Compute A^-1 for a 3×3 matrix, or find AX = B (which requires computing A^-1). Sometimes asks for the sum of entries or a specific entry.

Methods to Solve

Augmented matrix method: Row reduce [A | I] to [I | A^-1]
For 2×2: A^-1 = (1/det A) · [d -b; -c a]
Check det(A) ≠ 0 first; if det(A) = 0, write DNE
For AX = B: solve by computing X = A^-1B (row reduce [A | B])

5. Linear Transformations -- Standard Matrix Very High

MID 2020 Q6,Q8 • 2023 Q2,Q3 • 2024 Q2,Q4 | END 2019 Q5 • 2022 Q1 • 2023 Q1 | RES 2024 Q2

Find the standard matrix of a composed transformation: typically rotation (counter/clockwise) followed by reflection (across x-axis, y=-x, etc.), or shear. Also: given T on specific vectors, find T on another vector.

Methods to Solve

Rotation matrix (counter-clockwise by θ): [cosθ -sinθ; sinθ cosθ]
Clockwise rotation by θ: use -θ in the formula
Reflection across x-axis: [1 0; 0 -1]; y-axis: [-1 0; 0 1]; y=x: [0 1; 1 0]; y=-x: [0 -1; -1 0]
Reflection across x+y=0 (i.e., y=-x): [0 -1; -1 0]
Composition: If S comes after T, then standard matrix = [S][T] (right to left)
For "find T(v) given T on other vectors": express v as a linear combination of the known inputs, then use linearity

Example (Midterm 2024 Q4): Reflect through y=-x then rotate clockwise 120°. Compute [Rotation]·[Reflection].

6. Null Space Dimension / Rank-Nullity Theorem Very High

MID 2020 Q10 • 2023 Q9 • 2024 Q9 | END 2019 Q7 | RES 2019 Q2,Q4 • 2022 Q4

Given matrix dimensions and/or rank, find dim(Nul A). Or: for which parameter h does dim(Nul A) equal a specific value?

Methods to Solve

Rank-Nullity Theorem: rank(A) + dim(Nul A) = n (number of columns)
rank(A) = number of pivot columns after row reduction
dim(Nul A) = number of free variables = n - rank(A)
For parametric versions: row reduce with parameter, find when a pivot disappears

Example (Midterm 2020 Q10): A is 9×11 with rank 6. dim(Nul A) = 11 - 6 = 5.

7. Column Space Basis Identification High

MID 2020 Q12 | END 2022 Q5 • 2024 Q3 | RES 2019 Q3 • 2023 Q9

Given A row-equivalent to an echelon form U, determine which sets of original columns {a_i} form a basis for Col(A).

Methods to Solve

Pivot columns of A (columns corresponding to pivot positions in U) always form a basis for Col(A)
Any set of dim(Col A) linearly independent vectors from Col(A) is also a basis
Non-pivot columns are linear combinations of pivot columns, but can sometimes replace pivot columns in a basis
Standard basis vectors e_i are NOT guaranteed to be in Col(A) -- they form a basis for Col(A) only if A has full row rank
To check: a set of original columns is a basis iff the corresponding sub-matrix formed from U has full column rank

8. Coordinate Vectors w.r.t. a Basis High

MID 2020 Q13 • 2023 Q8 • 2024 Q8 | END 2022 Q6 | RES 2019 Q9,Q10 • 2022 Q6 • 2023 Q5 • 2024 Q9

Given a basis B = {b₁, b₂, ...} and a vector w, find [w]_B. Sometimes asks whether w lies in Span(B) at all.

Methods to Solve

Set up the equation c₁b₁ + c₂b₂ + ... = w
Row reduce the augmented matrix [b₁ b₂ ... | w]
If consistent: the solution gives the coordinate vector [w]_B
If inconsistent: w is not in Span(B), write DNE

9. Linear Independence / Dependence (with parameter) Medium

MID 2020 Q9 • 2024 Q3 | END 2019 Q2 | RES 2023 Q11

For which value(s) of α is a given set of vectors linearly dependent (or: has span dimension less than k)?

Methods to Solve

Form the matrix with the vectors as columns, row reduce
Vectors are linearly dependent iff det = 0 (for square matrices) or iff there's a free variable
For parametric: row reduce, find the parameter value that creates a zero row / eliminates a pivot
dim(Span) < k iff the vectors are linearly dependent

10. Eigenvalues and Eigenvectors Very High

MID 2020 Q11 | END 2019 Q11,Q14-15 • 2022 Q7-8 • 2023 Q5-7 • 2024 Q4,Q7 | RES 2019 Q11,Q13 • 2022 Q8 • 2023 Q4 • 2024 Q4,Q6

Find eigenvalues (including complex), eigenvectors, algebraic/geometric multiplicities. Given constraints on eigenvalues, determine properties of det(A) or invertibility.

Methods to Solve

Characteristic polynomial: det(A - λI) = 0
Eigenspace: E_λ = Nul(A - λI); find by row reducing (A - λI)
Algebraic multiplicity: multiplicity of λ as root of the characteristic polynomial
Geometric multiplicity: dim(E_λ) = dim Nul(A - λI)
For triangular matrices: eigenvalues are the diagonal entries
det(A) = product of eigenvalues; trace(A) = sum of eigenvalues
If Av = λBv for invertible A,B: multiply by A^-1 to get λ^-1 is eigenvalue of A^-1B

11. Diagonalizability (with parameter) High

END 2022 Q7 • 2023 Q7 • 2024 Q4 | RES 2019 Q12-13 • 2022 Q17

For which value of a parameter is a matrix diagonalizable? Check if geometric multiplicity equals algebraic multiplicity for each eigenvalue.

Methods to Solve

A is diagonalizable iff for every eigenvalue, geometric multiplicity = algebraic multiplicity
If all eigenvalues are distinct ⇒ automatically diagonalizable
For repeated eigenvalue λ with a.m. = k: check if dim Nul(A - λI) = k
Row reduce (A - λI) with the parameter to find when g.m. = a.m.
Symmetric matrices are always (orthogonally) diagonalizable

Example (Endterm 2022 Q7): A has eigenvalue 5 with a.m.=2. A is diagonalizable iff g.m.(5) = 2, which requires α = 1.

12. Discrete Dynamical Systems High

END 2022 Q8 • 2023 Q8 • 2024 Q2 | RES 2023 Q7 • 2024 Q8

Given x_k+1 = Ax_k, find the general solution, a specific component formula, or classify the origin (attractor / repeller / saddle).

Methods to Solve

Diagonalize A = PDP^-1, then x_k = PD^kP^-1x₀
General solution: x_k = c₁λ₁^kv₁ + c₂λ₂^kv₂
Use initial condition x₀ to solve for c₁, c₂
Origin classification:
- Attractor: all |λ_i| < 1
- Repeller: all |λ_i| > 1
- Saddle: mixed (one <1, one >1 in absolute value)
If one eigenvalue is 1: origin is neither attractor nor repeller

Example (Endterm 2024 Q2): A = [4 2; 1 5], eigenvectors [2;-1] and [1;1] with eigenvalues 3,6. General solution: c₁3^k[2;-1] + c₂6^k[1;1]. With x₀=[1;4]: y_n = 3·6ⁿ + 3ⁿ.

13. Least-Squares Solutions / Best Fit Line Very High

END 2019 Q21 • 2022 Q13-14 • 2023 Q10 • 2024 Q8 | RES 2019 Q19 • 2022 Q13 • 2023 Q10 • 2024 Q5

Find the least-squares solution of an inconsistent Ax = b, or the best-fit line y = a + bx through given data points.

Methods to Solve

Normal equations: A^TA x̂ = A^Tb
Compute A^TA and A^Tb, then solve the resulting system
If A^TA is singular: the LS solution has free variables (parametric)
Best-fit line: Set up A = [1 x₁; 1 x₂; ...], b = [y₁; y₂; ...], solve normal equations for [a; b]
LS error: ||b - A x̂|| = ||b - b̂||

Example (Endterm 2024 Q8): Points (1,10),(-1,-25),(3,10),(2,10). Set up design matrix, solve normal equations to get y = -10 + 9x.

14. Orthogonal Projection onto a Subspace Very High

END 2019 Q13 • 2022 Q10-11 • 2023 Q9 • 2024 Q5 | RES 2019 Q17 • 2022 Q12,Q14 • 2023 Q8 • 2024 Q10

Find proj_W(y) where W = Span{b₁, b₂}, or find the projection matrix, or compute the distance from y to W.

Methods to Solve

If {b₁, b₂} are orthogonal: proj_W(y) = (y·b₁/b₁·b₁)b₁ + (y·b₂/b₂·b₂)b₂
If not orthogonal: first apply Gram-Schmidt, or use the projection matrix P = A(A^TA)^-1A^T
Projection matrix: P = UU^T if U has orthonormal columns spanning W
Distance from y to W: ||y - proj_W(y)||
The standard matrix of the projection has P² = P and P^T = P

15. Gram-Schmidt Process Medium

END 2022 Q12 • 2024 Q5-6 | RES 2019 Q18

Apply Gram-Schmidt to a set of vectors to produce an orthogonal basis. Often asks for the third vector v₃ (up to rescaling).

Methods to Solve

v₁ = b₁
v₂ = b₂ - (b₂·v₁ / v₁·v₁)v₁
v₃ = b₃ - (b₃·v₁ / v₁·v₁)v₁ - (b₃·v₂ / v₂·v₂)v₂
Verify: each v_i should be orthogonal to all previous v_j (dot product = 0)
"Up to rescaling" means any scalar multiple of your answer is also correct

16. Orthogonal Complement (W^⊥) Medium

MID 2023 Q7 | END 2024 Q9 | RES 2022 Q15 • 2023 Q9 • 2024 Q7

Find a basis for W^⊥, or find a vector orthogonal to Col(A).

Methods to Solve

W^⊥ = Nul(A^T) where A has columns spanning W (or Col(A) = W)
If W is given by spanning vectors: form the matrix with those as rows, then find the null space
If W is given by an equation (e.g., 3x₂ + 2x₃ + 3x₄ = 0): the coefficient vector is the basis for W^⊥ (or find Nul of the constraint)
Vectors orthogonal to Col(A) lie in Nul(A^T), i.e., the left null space

17. Subspace Identification Medium

END 2019 Q6 • 2023 Q2-3

Determine whether a given subset of Rⁿ is a subspace. Find intersection of subspaces.

Methods to Solve

Check the three conditions: (1) contains zero vector, (2) closed under addition, (3) closed under scalar multiplication
Common non-subspaces: sets not containing 0, affine subsets (plane not through origin), unit vectors
Null spaces and column spaces of matrices are always subspaces
Intersection of subspaces: set up the system where both spanning representations are equal, solve

18. Determinant Properties (row ops, scalar mult, column swaps) Medium

MID 2024 Q6 | END 2023 Q5 | RES 2023 Q3

Given det(A) = x, compute det of a modified matrix (columns swapped, rows scaled, etc.).

Methods to Solve

Row swap: multiplies det by -1
Scaling a row by c: multiplies det by c
det(cA) = cⁿ det(A) for an n×n matrix
Column permutation: each transposition (swap of two columns) flips the sign
A cyclic permutation of 3 columns = 2 transpositions ⇒ no sign change
det(AB) = det(A)det(B)

19. Matrix Powers via Diagonalization Medium

END 2019 Q9 | RES 2019 Q15 • 2022 Q9-10 • 2024 Q3

Compute A^k or simplify A¹⁰⁰ using diagonalization or a recursive identity like A⁵ = -3A².

Methods to Solve

Diagonalization: A = PDP^-1 ⇒ A^k = PD^kP^-1
D^k is diagonal with entries λ_i^k
Recursive identity: If A⁵ = -3A², then A³ = -3I (when A is invertible), or repeatedly apply to reduce exponent
A¹⁰⁰ = (A³)³³ · A = (-3)³³A etc. — careful with the algebra
For rotation matrices: A^k = I when kθ = 2πm, i.e., k = 2π/θ

20. Complex Eigenvalues and A = PCP^-1 Decomposition Medium

END 2019 Q11 • 2022 Q9 • 2024 Q10 | RES 2019 Q14 • 2022 Q11 • 2023 Q6

Given a 2×2 matrix with complex eigenvalues a ± bi, express A = PCP^-1 where C = [a -b; b a]. Or: identify the rotation-scaling geometric interpretation.

Methods to Solve

Find eigenvalues a ± bi from the characteristic polynomial
Find an eigenvector v for eigenvalue a - bi (convention varies)
P = [Re(v) | Im(v)] and C = [a -b; b a]
Geometric interpretation: rotation by angle φ = arctan(b/a) scaled by r = √(a²+b²)
Matrix is "similar to" C means they share eigenvalues; (a,b) are uniquely determined (with b > 0)

21. Invertibility / det(A) from Matrix Equations High

MID 2020 Q1,Q11 • 2023 Q11 | END 2019 Q4 • 2022 Q1 • 2024 Q11 | RES 2019 Q5 • 2024 Q12

Given a matrix equation (e.g., A⁴(A^-1)^T = -2A^T or A³ = -5(A^T)^-1), find det(A). Also: for which α is A singular?

Methods to Solve

Take determinants of both sides: det(AB) = det(A)det(B), det(A^T) = det(A), det(A^-1) = 1/det(A)
det(cA) = cⁿdet(A) for n×n matrix
Solve the resulting polynomial equation for det(A)
For singularity: compute det(A) in terms of α and set = 0
If all entries of AB are strictly positive, both A and B must be invertible

Example (Endterm 2024 Q11): A³ = -5(A^T)^-1. Taking det: a³ = (-5)⁴/a, so a⁴ = 625, giving det(A) = ±5.

22. Range / Image of a Linear Transformation (with parameter) Medium

MID 2023 Q3 • 2024 Q2 | END 2024 Q1

For which value(s) of h does a vector b lie in the range (column space) of a transformation T?

Methods to Solve

b is in the range of T iff the system Ax = b is consistent (A is the standard matrix of T)
Row reduce [A | b] and check for contradictions
If parameterized: find values of h that avoid contradictory rows

23. LU Decomposition Low

RES 2019 Q7-8

Find the LU decomposition A = LU. Identify columns of L and U.

Methods to Solve

Row reduce A to echelon form U using only replacement operations (no row swaps)
L is lower triangular with 1s on diagonal; entries below diagonal are the negatives of the multipliers used
If row swaps are needed: use PA = LU (permuted LU)

24. Academic Reasoning (True/False Proofs) Very High

END 2019 Q22-25 • 2022 Q16-17 • 2023 Q11-12 • 2024 Q12-13 | RES 2019 Q20-22 • 2022 Q16-17 • 2023 Q11-12 • 2024 Q11-12

Prove a statement is true, or provide an explicit numerical counterexample to show it's false. Worth 3-5 points each. Appears on every endterm and resit (2 questions each).

Common Topics & Strategies

Frequent true statements:
- If v is eigenvector of A and B, then v is eigenvector of AB (prove by direct computation)
- ||u-v|| = ||u+v|| ⇒ u ⊥ v (expand dot products)
- Orthogonal projection eigenvalues are 0 or 1 (use P² = P)
- Matrix equation (e.g., 2A²+3A = 4I) implies invertibility (express A^-1 algebraically)
- If A^TA is diagonal, columns of A are orthogonal (use definition of A^TA entries)
Frequent false statements (need counterexample):
- A,B diagonalizable ⇒ AB diagonalizable (use simple 2×2 upper triangular)
- A row equivalent to B and B diagonalizable ⇒ A diagonalizable
- A similar to symmetric B ⇒ A orthogonally diagonalizable
- (A+B)(A-B) = A²-B² (fails because AB ≠ BA)
- λ=1 eigenvalue of A ⇒ λ=1 eigenvalue of row-equivalent B (false for λ≠0)
- v eigenvector of A² ⇒ v eigenvector of A
Proof strategies: direct computation, Invertible Matrix Theorem, properties of determinants, dot product expansion
Counterexample strategies: Try 2×2 matrices with simple entries (0,1); upper triangular matrices; diagonal matrices

25. Orthonormal Matrices and Properties Low

END 2019 Q19 • 2022 Q15 | RES 2019 Q16

Properties of orthogonal matrices (Q^TQ = I vs QQ^T = I), whether products of orthogonal matrices are orthogonal, etc.

Methods to Solve

Q has orthonormal columns iff Q^TQ = I
Q has orthonormal rows iff QQ^T = I
For square Q: Q^TQ = I ⇔ QQ^T = I ⇔ Q^-1 = Q^T
Product of orthogonal matrices is orthogonal: (UV)^T(UV) = V^TU^TUV = I
Sum of orthogonal matrices is NOT necessarily orthogonal (counterexample: I + (-I) = 0)

Generated from 11 exams: Midterms 2020, 2023, 2024 | Endterms 2019, 2022, 2023, 2024 | Resits 2019, 2022, 2023, 2024
CSE1205 Linear Algebra, TU Delft

CSE1205 Linear Algebra -- Exam Pattern Analysis

Table of Contents

1. Solving / Analyzing Linear Systems (with parameter) Very High

Methods to Solve

2. Matrix Equation Solving (Solve for X) Very High

Methods to Solve

3. Determinant Computation Very High

Methods to Solve

4. Matrix Inverse Computation High

Methods to Solve

5. Linear Transformations -- Standard Matrix Very High

Methods to Solve

6. Null Space Dimension / Rank-Nullity Theorem Very High

Methods to Solve

7. Column Space Basis Identification High

Methods to Solve

8. Coordinate Vectors w.r.t. a Basis High

Methods to Solve

9. Linear Independence / Dependence (with parameter) Medium

Methods to Solve

10. Eigenvalues and Eigenvectors Very High

Methods to Solve

11. Diagonalizability (with parameter) High

Methods to Solve

12. Discrete Dynamical Systems High

Methods to Solve

13. Least-Squares Solutions / Best Fit Line Very High

Methods to Solve

14. Orthogonal Projection onto a Subspace Very High

Methods to Solve

15. Gram-Schmidt Process Medium

Methods to Solve

16. Orthogonal Complement (W⊥) Medium

Methods to Solve

17. Subspace Identification Medium

Methods to Solve

18. Determinant Properties (row ops, scalar mult, column swaps) Medium

Methods to Solve

19. Matrix Powers via Diagonalization Medium

Methods to Solve

20. Complex Eigenvalues and A = PCP-1 Decomposition Medium

Methods to Solve

21. Invertibility / det(A) from Matrix Equations High

Methods to Solve

22. Range / Image of a Linear Transformation (with parameter) Medium

Methods to Solve

23. LU Decomposition Low

Methods to Solve

24. Academic Reasoning (True/False Proofs) Very High

Common Topics & Strategies

25. Orthonormal Matrices and Properties Low

Methods to Solve

16. Orthogonal Complement (W^⊥) Medium

20. Complex Eigenvalues and A = PCP^-1 Decomposition Medium