# 📐 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**
- Given: A = [[1, 2], [3, 4]], B = [[2, 0], [1, 3]]
- Calculate: AB = [[1·2+2·1, 1·0+2·3], [3·2+4·1, 3·0+4·3]]
- AB = [[4, 6], [10, 12]]
- Element (1,1) = 4
- **Points:** 10

**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**
- Given: A = [[1, 2], [3, 4]]
- Calculate: A^T = [[1, 3], [2, 4]] (rows become columns)
- **Points:** 10

---

### 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)**
- Given: A = [[2, 1], [4, 3]]
- Formula: A^(-1) = (1/det(A)) × [[d, -b], [-c, a]]
- det(A) = 2·3 - 1·4 = 2
- A^(-1) = (1/2) × [[3, -1], [-4, 2]]
- A^(-1) = [[3/2, -1/2], [-2, 1]]
- **Points:** 15

---

### 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**
- Given: A = [[1, 0, 0], [0, 2, 0], [0, 0, 3]]
- Trace = sum of diagonal elements
- tr(A) = 1 + 2 + 3 = 6
- **Points:** 5

---

## 📐 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
```bash
# 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)
```

---

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---

## 🎯 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