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📐 Linear Algebra Medium Exam Created

✅ Exam Details

Exam ID: linear-algebra-medium-v1 Title: Linear Algebra - Medium Level (Computational) Subject: Linear Algebra Difficulty: Intermediate (Medium) Duration: 50 minutes Passing Score: 70%

📊 Exam Structure

Total Questions: 10 (all require calculations) Total Points: 110 points Question Type: Single choice only (as requested)

Question Distribution

  • ✅ 10 Single Choice questions (110 points total)
  • ✅ All questions require calculations
  • ✅ "I Don't Know" option available

📚 Topics Covered & Calculations Required

Section 1: Vector Operations & Computations (3 questions, 30 points)

Q1: Dot Product Calculation

  • Given: u = [3, -2, 1], v = [1, 4, -2]
  • Calculate: u · v = (3)(1) + (-2)(4) + (1)(-2) = 3 - 8 - 2 = -3
  • Points: 10

Q2: Vector Magnitude

  • Given: w = [3, 4]
  • Calculate: ||w|| = √(3² + 4²) = √(9 + 16) = √25 = 5
  • Points: 10

Q3: Unit Vector

  • Given: v = [6, 8]
  • Calculate: ||v|| = √(36 + 64) = √100 = 10
  • Unit vector: v/||v|| = [6/10, 8/10] = [3/5, 4/5]
  • Points: 10

Section 2: Matrix Operations & Calculations (3 questions, 30 points)

Q4: Matrix Multiplication

Q5: Determinant (2×2)

  • Given: A = 2, 1], [4, 3
  • Calculate: det(A) = (2)(3) - (1)(4) = 6 - 4 = 2
  • Points: 10

Q6: Matrix Transpose

Section 3: Linear Systems & Solutions (2 questions, 30 points)

Q7: Solve 2×2 System

  • System: x + 2y = 7, 2x - y = 4
  • Method 1 (Substitution):
    • From eq1: x = 7 - 2y
    • Sub into eq2: 2(7 - 2y) - y = 4
    • 14 - 4y - y = 4
    • -5y = -10
    • y = 2
    • x = 7 - 2(2) = 3
  • Answer: x = 3, y = 2
  • Points: 15

Q8: Matrix Inverse (2×2)

Section 4: Eigenvalues & Special Computations (2 questions, 20 points)

Q9: Eigenvalues

  • Given: A = 3, 1], [1, 3
  • Characteristic equation: det(A - λI) = 0
  • det(3-λ, 1], [1, 3-λ) = 0
  • (3-λ)(3-λ) - (1)(1) = 0
  • (3-λ)² - 1 = 0
  • 9 - 6λ + λ² - 1 = 0
  • λ² - 6λ + 8 = 0
  • (λ - 4)(λ - 2) = 0
  • λ₁ = 4, λ₂ = 2
  • Points: 15

Q10: Trace of Matrix

📐 Calculation Techniques Tested

1. Vector Operations

  • Dot product computation
  • Vector magnitude (Euclidean norm)
  • Normalization (unit vectors)

2. Matrix Arithmetic

  • Matrix multiplication (row × column)
  • Transpose operation
  • Determinant (2×2 formula)

3. Linear Systems

  • System of equations solving
  • Substitution or elimination method
  • Matrix inverse formula (2×2)

4. Eigenvalues

  • Characteristic equation
  • Solving quadratic equations
  • Determinant of (A - λI)

5. Matrix Properties

  • Trace (diagonal sum)
  • Special matrix identification

✨ Features Enabled

✅ "I Don't Know" Option

  • Available on all questions
  • Encourages honest self-assessment
  • Scores 0 points (no penalty)

✅ Automatic Scoring

  • All questions are single choice
  • Immediate results upon submission
  • No manual grading required

✅ Computational Focus

  • All questions require calculations
  • Tests practical problem-solving
  • Verifies understanding through computation

🎓 Learning Objectives

This exam assesses ability to:

  1. Perform Vector Operations

    • Compute dot products accurately
    • Calculate vector magnitudes
    • Find unit vectors through normalization

  2. Execute Matrix Calculations

    • Multiply matrices correctly
    • Transpose matrices
    • Compute 2×2 determinants

  3. Solve Linear Systems

    • Use algebraic methods (substitution/elimination)
    • Find matrix inverses for 2×2 matrices
    • Apply inverse formula correctly

  4. Work with Eigenvalues

    • Set up characteristic equations
    • Solve for eigenvalues
    • Understand matrix properties (trace)

🎯 Difficulty Level: Medium

Why Medium Difficulty?

Computational Requirements:

  • Multi-step calculations
  • Matrix multiplication (not trivial)
  • System solving (2 equations, 2 unknowns)
  • Eigenvalue computation (requires polynomial solving)
  • Matrix inverse (requires formula knowledge)

Knowledge Requirements:

  • Understand formulas, not just plug-and-chug
  • Know when to apply which technique
  • Interpret results correctly

Not Beginner Because:

  • Requires matrix multiplication mastery
  • Needs eigenvalue concepts
  • Matrix inverse is non-trivial
  • Multi-step problem solving

Not Advanced Because:

  • Only 2×2 and 3×3 matrices
  • No abstract theory questions
  • No proofs required
  • Standard computational problems

📊 Expected Performance

Score Ranges

  • 90-100% - Excellent computational skills
  • 80-89% - Strong understanding, minor calculation errors
  • 70-79% - Passing, review some concepts
  • 60-69% - Below passing, need more practice
  • <60% - Need to study fundamentals more

Time Management

  • Q1-Q3 (Vectors): 2-3 minutes each
  • Q4-Q6 (Matrices): 3-4 minutes each
  • Q7-Q8 (Systems): 5-7 minutes each
  • Q9 (Eigenvalues): 6-8 minutes
  • Q10 (Trace): 1-2 minutes
  • Total estimate: 35-40 minutes + 10 min review

🔗 Integration with Learning Plan

This exam aligns with:

  • Linear Algebra Learning Plan - Phase 1-2
  • Covers Modules 1.1-1.3, 2.1-2.3, 2.6
  • Tests computational fluency
  • Prepares for Phase 3 (advanced decompositions)

Recommended Study Before Taking

  1. Complete Linear Algebra Modules 1.1-1.3 (Foundations)
  2. Complete Modules 2.1-2.3 (Systems, Inverses, Determinants)
  3. Study Module 2.6 (Eigenvalues basics)
  4. Practice 50+ similar problems
  5. Implement in Python/NumPy

After Passing

  1. Review any weak areas
  2. Continue to Module 2.4 (Vector Spaces)
  3. Study Module 3.3 (Matrix Decompositions)
  4. Take advanced exam (future)

🚀 How to Access

Via Web Interface

  1. Go to http://localhost
  2. Login or register
  3. Navigate to "Available Exams"
  4. Select "Linear Algebra - Medium Level (Computational)"
  5. Click "Start Exam"

Via API

# List all exams
curl http://localhost/api/exams/

# Get Linear Algebra exam details
curl http://localhost/api/exams/linear-algebra-medium-v1/

# Start attempt (requires authentication)
curl -X POST http://localhost/api/exams/linear-algebra-medium-v1/start

📝 Calculation Reference

Quick Formulas Needed

Dot Product:

a · b = a₁b₁ + a₂b₂ + a₃b₃

Vector Magnitude:

||v|| = √(v₁² + v₂² + v₃²)

Unit Vector:

û = v / ||v||

Matrix Multiplication (2×2):

[[a, b], [c, d]] × [[e, f], [g, h]] = [[ae+bg, af+bh], [ce+dg, cf+dh]]

Determinant (2×2):

det([[a, b], [c, d]]) = ad - bc

Matrix Inverse (2×2):

A^(-1) = (1/det(A)) × [[d, -b], [-c, a]]

Eigenvalues (2×2):

det(A - λI) = 0
Solve characteristic polynomial

Trace:

tr(A) = Σ Aᵢᵢ (sum of diagonal)

📈 Available Exams in System

  1. Python Easy (10 questions)
  2. Python Easy 15Q (15 questions)
  3. Python Intermediate (50 questions)
  4. C++ Easy (20 questions)
  5. Linear Algebra Medium (10 questions) ⭐ NEW

🎯 Recommended Preparation

Tools Needed

  • Paper and pencil for calculations
  • OR Python + NumPy to verify
  • Calculator (optional)

Practice Problems

Before taking this exam, practice:

  • 20+ dot product problems
  • 20+ matrix multiplication problems
  • 10+ determinant calculations
  • 10+ system solving problems
  • 5+ eigenvalue problems

Study Resources

  • Linear Algebra Learning Plan: Modules 1.1-1.3, 2.1-2.3, 2.6
  • 3Blue1Brown videos 1-5
  • Gilbert Strang lectures 1-10
  • Practice with NumPy to verify calculations

🌟 Exam Features

Computational Focus

✅ Real calculations required ✅ Multi-step problem solving ✅ Tests practical skills ✅ Verifies formula knowledge

Auto-Grading

✅ Single choice format ✅ Immediate results ✅ Detailed feedback ✅ Score breakdown by section

Student-Friendly

✅ "I don't know" option available ✅ Auto-save every 10 seconds ✅ Timer with warnings ✅ Navigation between questions

Your Linear Algebra medium exam is ready! Test your computational skills! 📐✨

Created: October 21, 2025 Status: ✅ Published and Available Access: http://localhost