MATH 2220
Linear Algebra with Applications
Complete Study Guide — All 13 Chapters
Independent Study Resource
Not affiliated with or endorsed by Yale University
Aligned with the Yale MATH 2220 syllabus
Edition 2026
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,Table of Contents
1. Vectors and Linear Combinations
2. Solving Linear Systems and Gaussian Elimination
3. Matrix Algebra: Inverses, Transposes, and Permutations
4. Vector Spaces and Subspaces
5. Linear Independence, Basis, and Dimension
6. Nullspace, Column Space, and Rank
7. Linear Transformations
8. Orthogonality, Projections, and Least Squares
9. Gram-Schmidt and QR Decomposition
10. Determinants
11. Eigenvalues and Eigenvectors
12. Diagonalization and Similar Matrices
13. Singular Value Decomposition and Applications
14. Glossary of Key Terms
15. Formula Sheet
16. Final Exam Strategy and Study Checklist
17. References and Attribution
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, Chapter 1 — Vectors and Linear Combinations
Learning Objectives
1. Define vectors in Rⁿ and perform vector operations.
2. Compute dot products and norms.
3. Express vectors as linear combinations.
4. Understand the geometric interpretation of vectors.
Key Definitions
Term Definition
Vector An ordered list of numbers: v = (v₁, v₂, ..., vₙ).
Vector Addition Component-wise addition: (u₁+v₁, ..., uₙ+vₙ).
Scalar Multiplication c·v = (cv₁, cv₂, ..., cvₙ).
Dot Product u·v = u₁v₁ + u₂v₂ + ... + uₙvₙ.
Norm (Length) ||v|| = √(v·v) = √(v₁² + ... + vₙ²).
Linear Combination c₁v₁ + c₂v₂ + ... + cₖvₖ for scalars cᵢ.
1.1 Vector Operations
u + v = (u₁+v₁, u₂+v₂, ..., uₙ+vₙ)
c·u = (cu₁, cu₂, ..., cuₙ)
u·v = u₁v₁ + u₂v₂ + ... + uₙvₙ
||v|| = √(v·v)
1.2 Linear Combinations
A vector b is a linear combination of vectors v₁, ..., vₖ if there exist scalars c₁, ..., cₖ such that b = c₁v₁ +
... + cₖvₖ. This is equivalent to asking whether the system with augmented matrix [v₁ ... vₖ | b] is
consistent.
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Linear Algebra with Applications
Complete Study Guide — All 13 Chapters
Independent Study Resource
Not affiliated with or endorsed by Yale University
Aligned with the Yale MATH 2220 syllabus
Edition 2026
Explore our developer-friendly HTML to PDF API Printed using PDFCrowd HTML to PDF
,Table of Contents
1. Vectors and Linear Combinations
2. Solving Linear Systems and Gaussian Elimination
3. Matrix Algebra: Inverses, Transposes, and Permutations
4. Vector Spaces and Subspaces
5. Linear Independence, Basis, and Dimension
6. Nullspace, Column Space, and Rank
7. Linear Transformations
8. Orthogonality, Projections, and Least Squares
9. Gram-Schmidt and QR Decomposition
10. Determinants
11. Eigenvalues and Eigenvectors
12. Diagonalization and Similar Matrices
13. Singular Value Decomposition and Applications
14. Glossary of Key Terms
15. Formula Sheet
16. Final Exam Strategy and Study Checklist
17. References and Attribution
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, Chapter 1 — Vectors and Linear Combinations
Learning Objectives
1. Define vectors in Rⁿ and perform vector operations.
2. Compute dot products and norms.
3. Express vectors as linear combinations.
4. Understand the geometric interpretation of vectors.
Key Definitions
Term Definition
Vector An ordered list of numbers: v = (v₁, v₂, ..., vₙ).
Vector Addition Component-wise addition: (u₁+v₁, ..., uₙ+vₙ).
Scalar Multiplication c·v = (cv₁, cv₂, ..., cvₙ).
Dot Product u·v = u₁v₁ + u₂v₂ + ... + uₙvₙ.
Norm (Length) ||v|| = √(v·v) = √(v₁² + ... + vₙ²).
Linear Combination c₁v₁ + c₂v₂ + ... + cₖvₖ for scalars cᵢ.
1.1 Vector Operations
u + v = (u₁+v₁, u₂+v₂, ..., uₙ+vₙ)
c·u = (cu₁, cu₂, ..., cuₙ)
u·v = u₁v₁ + u₂v₂ + ... + uₙvₙ
||v|| = √(v·v)
1.2 Linear Combinations
A vector b is a linear combination of vectors v₁, ..., vₖ if there exist scalars c₁, ..., cₖ such that b = c₁v₁ +
... + cₖvₖ. This is equivalent to asking whether the system with augmented matrix [v₁ ... vₖ | b] is
consistent.
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