# Linear Algebra Initial Assessment

## 🎯 Purpose

This assessment will help determine your current Linear Algebra proficiency level and create a personalized learning path.

## 📋 Assessment Structure

### Part 1: Self-Assessment Questionnaire
### Part 2: Computational Problems
### Part 3: Knowledge Gap Analysis

---

## Part 1: Self-Assessment Questionnaire

Rate yourself honestly (0-4):
- **Level 0:** Never heard of it
- **Level 1:** Basic awareness
- **Level 2:** Can use with reference
- **Level 3:** Proficient, confident
- **Level 4:** Expert, can teach

### Vectors
| Topic | Level (0-4) | Notes |
|-------|-------------|-------|
| Vector definition & notation | | |
| Vector addition/subtraction | | |
| Scalar multiplication | | |
| Dot product | | |
| Cross product (3D) | | |
| Vector magnitude/norm | | |
| Unit vectors | | |
| Orthogonal vectors | | |
| Vector projection | | |
| Linear combinations | | |

### Matrices
| Topic | Level (0-4) | Notes |
|-------|-------------|-------|
| Matrix definition | | |
| Matrix addition/subtraction | | |
| Matrix multiplication | | |
| Transpose | | |
| Identity matrix | | |
| Inverse matrix | | |
| Determinants | | |
| Special matrices (diagonal, symmetric) | | |

### Linear Systems
| Topic | Level (0-4) | Notes |
|-------|-------------|-------|
| Systems of linear equations | | |
| Gaussian elimination | | |
| Row echelon form | | |
| RREF | | |
| Solution types (unique, infinite, none) | | |
| Homogeneous systems | | |
| Augmented matrices | | |

### Vector Spaces
| Topic | Level (0-4) | Notes |
|-------|-------------|-------|
| Vector space definition | | |
| Subspaces | | |
| Span | | |
| Linear independence | | |
| Basis | | |
| Dimension | | |
| Null space | | |
| Column space | | |
| Rank | | |

### Eigenvalues & Decompositions
| Topic | Level (0-4) | Notes |
|-------|-------------|-------|
| Eigenvalues | | |
| Eigenvectors | | |
| Characteristic polynomial | | |
| Diagonalization | | |
| LU decomposition | | |
| QR decomposition | | |
| SVD | | |
| Orthogonalization (Gram-Schmidt) | | |

### Applications
| Topic | Level (0-4) | Notes |
|-------|-------------|-------|
| Linear regression | | |
| PCA | | |
| Graphics transformations | | |
| Least squares | | |
| Optimization | | |

---

## Part 2: Computational Problems

### Problem 1: Vector Operations (Beginner)
Given vectors u = [2, -1, 3] and v = [1, 4, -2]:

a) Compute u + v
b) Compute 3u - 2v  
c) Compute ||u|| (magnitude)
d) Compute u · v (dot product)
e) Are u and v orthogonal?

**Can you solve this?** ☐ Yes ☐ No ☐ Partially

---

### Problem 2: Matrix Multiplication (Beginner)
Compute AB where:
```
A = [1  2]    B = [5  6]
    [3  4]        [7  8]
```

**Can you solve this?** ☐ Yes ☐ No ☐ With formula

---

### Problem 3: Solve Linear System (Intermediate)
Solve using Gaussian elimination:
```
 x + 2y - z = 3
2x - y + z = 1
3x + y + 2z = 11
```

**Can you solve this?** ☐ Yes ☐ No ☐ With steps

---

### Problem 4: Matrix Inverse (Intermediate)
Find the inverse of:
```
A = [2  1]
    [5  3]
```

**Can you solve this?** ☐ Yes ☐ No ☐ With formula

---

### Problem 5: Eigenvalues (Advanced)
Find eigenvalues and eigenvectors of:
```
A = [3  1]
    [1  3]
```

**Can you solve this?** ☐ Yes ☐ No ☐ With steps

---

### Problem 6: Application - Linear Regression (Advanced)
Given data points: (1,2), (2,4), (3,5), (4,6)

Set up and solve the least squares problem to find the best-fit line y = mx + b using matrix methods.

**Can you solve this?** ☐ Yes ☐ No ☐ Know concept only

---

## Part 3: Knowledge Gap Analysis

### Based on Self-Assessment

**Count your scores:**
- Topics at Level 0: ___
- Topics at Level 1: ___
- Topics at Level 2: ___
- Topics at Level 3: ___
- Topics at Level 4: ___

**Total topics:** ___

### Based on Problems

**Problems solved:**
- Problem 1 (Vectors): ☐
- Problem 2 (Matrix Mult): ☐
- Problem 3 (Systems): ☐
- Problem 4 (Inverse): ☐
- Problem 5 (Eigenvalues): ☐
- Problem 6 (Application): ☐

**Total solved:** ___ / 6

---

## 📊 Proficiency Level Determination

### Absolute Beginner (0-20% Level 2+, 0-1 problems)
- **Start:** Phase 1 from Module 1.1
- **Timeline:** 10-12 months to applications
- **Focus:** Build from scratch, emphasize geometric intuition
- **Resources:** 3Blue1Brown, Khan Academy, "Linear Algebra Done Right"

### Beginner (20-40% Level 2+, 1-2 problems)
- **Start:** Phase 1 with quick review, focus on Phase 2
- **Timeline:** 8-10 months to applications
- **Focus:** Strengthen basics, master systems and inverses
- **Resources:** Gilbert Strang lectures, "Linear Algebra and Its Applications"

### Intermediate (40-60% Level 2+, 3-4 problems)
- **Start:** Phase 2, review Phase 1 as needed
- **Timeline:** 6-8 months to applications
- **Focus:** Vector spaces, eigenvalues, decompositions
- **Resources:** Strang's book, MIT OCW

### Advanced (60-80% Level 2+, 5 problems)
- **Start:** Phase 3, skim Phase 1-2
- **Timeline:** 4-6 months to specialization
- **Focus:** Advanced theory and applications
- **Resources:** "Matrix Analysis", research papers

### Expert (80%+ Level 3+, 6 problems)
- **Start:** Phase 4-5 (Applications & Specialization)
- **Timeline:** 2-4 months to deep specialization
- **Focus:** Specialized applications, cutting-edge topics
- **Resources:** Research papers, advanced texts

---

## 🎯 Personalized Learning Path

### Your Starting Point
**Based on assessment:** _______________ 

### Recommended Phase
**Start at Phase:** _______________

### Topics to Review First
1. _______________
2. _______________
3. _______________

### Topics to Skip (Already Mastered)
1. _______________
2. _______________

### Weak Areas to Focus On
1. _______________
2. _______________

### Estimated Timeline to Advanced
**From your starting point:** ___ months

---

## 📝 Action Items

### Immediate (This Week)
1. ☐ Complete this assessment
2. ☐ Set up Python + NumPy or MATLAB
3. ☐ Watch 3Blue1Brown: "Essence of Linear Algebra" (video 1)
4. ☐ Review recommended phase in Master Plan
5. ☐ Join math communities (r/learnmath, Math Stack Exchange)

### First Month
1. ☐ Complete ____ modules
2. ☐ Solve 100+ practice problems
3. ☐ Watch all 3Blue1Brown videos (11 total)
4. ☐ Implement basic operations in code
5. ☐ Take first monthly exam

---

## 🔄 Reassessment Schedule

- **Week 4:** Quick progress check
- **Month 3:** Comprehensive reassessment
- **Month 6:** Mid-journey assessment
- **Month 9:** Full reassessment
- **Month 12:** Expert level check

---

## 📚 Additional Resources

### Video Series
- **3Blue1Brown:** "Essence of Linear Algebra" (MUST WATCH)
- **MIT OCW:** Gilbert Strang's 18.06
- **Khan Academy:** Linear Algebra playlist

### Interactive Tools
- **GeoGebra:** Visualize vectors and transformations
- **WolframAlpha:** Compute anything
- **MATLAB/Octave:** Numerical experiments
- **Python + NumPy:** Programming practice

### Problem Sources
- MIT OCW problem sets
- Gilbert Strang's textbook exercises
- Linear Algebra Done Right exercises
- Math Stack Exchange

---

**Date Completed:** _______________
**Next Reassessment:** _______________
**Notes:**
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