lnet_tutor/learning_plans/linear_algebra/02_INITIAL_ASSESSMENT.md

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

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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

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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

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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

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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

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Problem 4: Matrix Inverse (Intermediate)

Find the inverse of:

A = [2  1]
    [5  3]

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

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Problem 5: Eigenvalues (Advanced)

Find eigenvalues and eigenvectors of:

A = [3  1]
    [1  3]

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

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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

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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

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📊 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

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🎯 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

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📝 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

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🔄 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

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📚 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

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Date Completed: _______________ Next Reassessment: _______________ Notes:

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