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# Linear Algebra

Dive deep into the math behind vectors and matrices. Learn a wide assortment of computational methods and conceptual connections that unify into an elegant whole.

## Content

### Determinants

• Compute the determinant of an NxN matrix using Laplace expansions.
• Understand the effect of row/column operations on the determinant and use them to simplify determinant computations.
• Use Cramer’s rule to solve linear systems.

### Gaussian Elimination

• Use Gaussian elimination to solve systems of linear equations and compute inverse matrices.
• Identify the pivot columns and rank of a matrix by converting to row echelon form.
• Find an LU factorization for a given matrix and apply LU factorization to solve linear systems.
• Construct the transition matrix for a Markov chain and find the associated steady-state vector.

### Subspaces and Dimension

• Understand span and linear independence in both algebraic and geometric contexts.
• Determine whether a set of vectors represents a subspace, and if so, find a basis and identify its dimension.
• Compute the column space and nullspace of a matrix.
• Express coordinates of vectors relative to different bases.
• Generalize concepts of subspaces and dimension to abstract vector spaces.

### Eigenvalues, Eigenvectors, and Diagonalization

• Understand eigenvalues/eigenvectors geometrically and calculate them algebraically.
• Find the characteristic equation of a matrix and apply the Cayley-Hamilton Theorem to evaluate matrix polynomials.
• Diagonalize matrices, use diagonalization to compute large powers of matrices, and interpret conclusions of the spectral theorem.
• Construct rotation-rescaling matrices and interpret matrices with complex eigenvalues as representing rotation-rescaling matrices.
• Compute generalized eigenvectors and Jordan canonical decomposition for non-diagonalizable matrices.

### Orthogonality

• Identify orthogonal vectors, compute the orthogonal complement of a subspace, and understand orthogonality relationships between the four fundamental subspaces.
• Project a vector onto a subspace and apply subspace projection to fit various regression models to data.
• Apply the Gram-Schmidt process to find an orthogonal basis for a given subspace.
• Find a QR factorization for a given matrix and apply QR factorization to solve linear systems.
• Compute and interpret properties of quadratic forms including definiteness and principal axes.
• Find the singular value decomposition of a given matrix and identify/interpret the principal components.
1.
Preliminaries
21 topics
1.1. Sets
 1.1.1. Special Sets 1.1.2. Set-Builder Notation 1.1.3. Equivalent Sets 1.1.4. Cardinality of Sets 1.1.5. Subsets 1.1.6. The Complement of a Set 1.1.7. The Difference of Sets 1.1.8. The Cartesian Product
1.2. Vector Geometry
 1.2.1. The Vector Equation of a Line 1.2.2. The Parametric Equations of a Line 1.2.3. Writing the Cartesian Equation of a Line from the Vector Equation 1.2.4. Finding the Vector Equation of a Plane Using the Dot Product 1.2.5. The Cartesian Equation of a Plane 1.2.6. The Parametric Equations of a Plane
1.3. Logical Quantifiers
 1.3.1. Statements and Propositions 1.3.2. Universal and Existential Quantifiers 1.3.3. Formal and Informal Language
1.4. Functions
 1.4.1. Sets and Functions 1.4.2. Surjections 1.4.3. Injections 1.4.4. Bijections
2.
Matrices
36 topics
2.5. Determinants
 2.5.1. Cramer’s Rule for 2x2 Systems of Linear Equations 2.5.2. Cramer’s Rule for 3x3 Systems 2.5.3. The Determinant of a NxN Matrix 2.5.4. Finding Determinants Using Laplace Expansions 2.5.5. Basic Properties of Determinants 2.5.6. Further Properties of Determinants 2.5.7. Row and Column Operations on Determinants 2.5.8. Conditions When a Determinant Equals Zero 2.5.9. Finding Determinants Using Row and Column Operations 2.5.10. Partitioned and Block Matrices
2.6. Gaussian Elimination
 2.6.1. Systems of Equations as Augmented Matrices 2.6.2. Row Echelon Form 2.6.3. Solving Systems of Equations Using Back Substitution 2.6.4. Elementary Row Operations 2.6.5. Creating Rows or Columns Containing Zeros Using Gaussian Elimination 2.6.6. Solving 2x2 Systems of Equations Using Gaussian Elimination 2.6.7. Solving 2x2 Singular Systems of Equations Using Gaussian Elimination 2.6.8. Solving 3x3 Systems of Equations Using Gaussian Elimination 2.6.9. Identifying the Pivot Columns of a Matrix 2.6.10. Solving 3x3 Singular Systems of Equations Using Gaussian Elimination 2.6.11. Reduced Row Echelon Form 2.6.12. Gaussian Elimination For NxM Systems of Equations
2.7. Elementary Matrices
 2.7.1. Elementary 2x2 Matrices 2.7.2. Row Operations on 2x2 Matrices as Products of Elementary Matrices 2.7.3. Elementary 3x3 Matrices 2.7.4. Row Operations on 3x3 Matrices as Products of Elementary Matrices
2.8. The Inverse of a Matrix
 2.8.1. The Invertible Matrix Theorem in Terms of 2x2 Systems of Equations 2.8.2. Finding the Inverse of a 2x2 Matrix Using Row Operations 2.8.3. Finding the Inverse of a 3x3 Matrix Using Row Operations 2.8.4. Finding the Inverse of an NxN Square Matrix Using Row Operations 2.8.5. Matrices With Easy-to-Find Inverses
2.9. LU Factorization
 2.9.1. Triangular Matrices 2.9.2. LU Factorization of 2x2 Matrices 2.9.3. LU Factorization of 3x3 Matrices 2.9.4. LU Factorization of NxN Matrices 2.9.5. Solving Systems of Equations Using LU Factorization
3.
Vector Spaces
26 topics
3.10. Vectors in N-Dimensional Space
 3.10.1. Vectors in N-Dimensional Euclidean Space 3.10.2. Linear Combinations of Vectors in N-Dimensional Euclidean Space 3.10.3. Linear Span of Vectors in N-Dimensional Euclidean Space 3.10.4. Linear Dependence and Independence
3.11. Subspaces of N-Dimensional Space
 3.11.1. Subspaces of N-Dimensional Space 3.11.2. The Column Space of a Matrix 3.11.3. The Null Space of a Matrix
3.12. Bases of N-Dimensional Space
 3.12.1. Finding a Basis of a Span 3.12.2. Finding a Basis of the Column Space of a Matrix 3.12.3. Finding a Basis of the Null Space of a Matrix 3.12.4. Expressing the Coordinates of a Vector in a Given Basis 3.12.5. Writing Vectors in Different Bases 3.12.6. The Change-of-Coordinates Matrix 3.12.7. Changing a Basis Using the Change-of-Coordinates Matrix
3.13. Dimension and Rank in N-Dimensional Space
 3.13.1. The Dimension of a Span 3.13.2. The Rank of a Matrix 3.13.3. The Dimension of the Null Space of a Matrix 3.13.4. The Invertible Matrix Theorem in Terms of Dimension, Rank and Nullity 3.13.5. The Rank-Nullity Theorem
3.14. Abstract Vector Spaces
 3.14.1. Introduction to Abstract Vector Spaces 3.14.2. Defining Abstract Vector Spaces 3.14.3. Linear Independence in Abstract Vector Spaces 3.14.4. Subspaces of Abstract Vector Spaces 3.14.5. Bases in Abstract Vector Spaces 3.14.6. The Coordinates of a Vector Relative to a Basis in Abstract Vector Spaces 3.14.7. Dimension in Abstract Vector Spaces
4.
Linear Transformations
11 topics
4.15. Linear Transformations
 4.15.1. The Standard Matrix of a Linear Transformation in Terms of the Standard Basis 4.15.2. The Kernel of a Linear Transformation 4.15.3. The Image and Rank of a Linear Transformation 4.15.4. Singular Linear Transformations in the Cartesian Plane 4.15.5. Finding the Image of a Linear Transformation in the Cartesian Plane 4.15.6. The Invertible Matrix Theorem in Terms of Linear Transformations 4.15.7. The Rank-Nullity Theorem in Terms of Linear Transformations
4.16. Linear Maps and Their Matrix Representations
 4.16.1. The Matrix of a Linear Transformation Relative to a Basis 4.16.2. Connections Between Matrix Representations of a Linear Transformation 4.16.3. Linear Maps Between Two Different Vector Spaces 4.16.4. The Matrix of a Linear Map Relative to Two Bases
5.
Diagonalization of Matrices
28 topics
5.17. Eigenvectors and Eigenvalues
 5.17.1. The Eigenvalues and Eigenvectors of a 2x2 Matrix 5.17.2. Calculating the Eigenvalues of a 2x2 Matrix 5.17.3. Calculating the Eigenvectors of a 2x2 Matrix 5.17.4. The Characteristic Equation of a Matrix 5.17.5. The Cayley-Hamilton Theorem and Its Applications 5.17.6. Calculating the Eigenvectors of a 3x3 Matrix With Distinct Eigenvalues 5.17.7. Calculating the Eigenvectors of a 3x3 Matrix in the General Case 5.17.8. The Invertible Matrix Theorem in Terms of Eigenvalues
5.18. Diagonalization
 5.18.1. Diagonalizing a 2x2 Matrix 5.18.2. Properties of Diagonalization 5.18.3. Diagonalizing a 3x3 Matrix With Distinct Eigenvalues 5.18.4. Diagonalizing a 3x3 Matrix in the General Case
5.19. Real Matrices With Complex Eigenvalues
 5.19.1. Vectors Over the Complex Numbers 5.19.2. Matrices Over the Complex Numbers 5.19.3. Finding Complex Eigenvalues of Real 2x2 Matrices 5.19.4. Finding Complex Eigenvectors of Real 2x2 Matrices 5.19.5. Rotation-Scaling Matrices 5.19.6. Reducing Real 2x2 Matrices to Rotation-Scaling Form 5.19.7. Block Diagonalization of NxN Matrices
5.20. Generalized Eigenvectors
 5.20.1. Nilpotent and Idempotent Matrices 5.20.2. Generalized Eigenvectors 5.20.3. Ranks of Generalized Eigenvectors 5.20.4. Finding Generalized Eigenvectors of Specific Ranks
5.21. Jordan Canonical Decomposition
 5.21.1. Jordan Blocks and Jordan Matrices 5.21.2. Jordan Canonical Form of a 2x2 Matrix 5.21.3. Jordan Canonical Decomposition of a 2x2 Matrix 5.21.4. Jordan Canonical Form of a 3x3 Matrix 5.21.5. Jordan Canonical Decomposition of a 3x3 Matrix
6.
Projections
24 topics
6.22. Inner Products
 6.22.1. The Dot Product in N-Dimensional Euclidean Space 6.22.2. The Norm of a Vector in N-Dimensional Euclidean Space 6.22.3. Inner Product Spaces 6.22.4. The Inner Product in Vector Spaces Over the Complex Numbers 6.22.5. The Norm of a Vector in Inner Product Spaces
6.23. Orthogonality
 6.23.1. Orthogonal Vectors in Euclidean Spaces 6.23.2. Orthogonal Vectors in Inner Product Spaces 6.23.3. The Cauchy-Schwarz Inequality and the Angle Between Two Vectors 6.23.4. The Pythagorean Theorem and the Triangle Inequality 6.23.5. Orthogonal Complements 6.23.6. Orthogonal Sets in Euclidean Spaces 6.23.7. Orthogonal Sets in Inner Product Spaces 6.23.8. Orthogonal Matrices and Linear Transformations 6.23.9. The Four Fundamental Subspaces of a Matrix
6.24. Orthogonal Projections
 6.24.1. Projecting Vectors Onto One-Dimensional Subspaces 6.24.2. The Components of a Vector with Respect to an Orthogonal or Orthonormal Basis 6.24.3. Projecting Vectors Onto Subspaces in Euclidean Spaces (Orthogonal Bases) 6.24.4. Projecting Vectors Onto Subspaces in Euclidean Spaces (Arbitrary Bases) 6.24.5. Projecting Vectors Onto Subspaces in Euclidean Spaces (Arbitrary Bases): Applications 6.24.6. Projection Matrices, Linear Transformations and Their Properties 6.24.7. Projecting Vectors Onto Subspaces in Inner Product Spaces
6.25. Orthogonalization Processes
 6.25.1. The Gram-Schmidt Process for Two Vectors 6.25.2. The Gram-Schmidt Process in the General Case 6.25.3. QR Factorization
7.
19 topics
7.26. Diagonalization of Symmetric Matrices
 7.26.1. Symmetric Matrices 7.26.2. Diagonalization of 2x2 Symmetric Matrices 7.26.3. Diagonalization of 3x3 Symmetric Matrices 7.26.4. The Spectral Theorem
 7.27.1. Bilinear Forms 7.27.2. Quadratic Forms 7.27.3. Change of Variables in Quadratic Forms 7.27.4. Finding the Canonical Form of a Quadratic Form Using Lagrange's Method 7.27.5. Finding the Canonical Form of a Quadratic Form Using Orthogonal Transformations 7.27.6. Positive-Definite and Negative-Definite Quadratic Forms
7.28. Quadratic Forms in Euclidian Space
 7.28.1. Reducing a Quadratic Curve to Its Principal Axes 7.28.2. Classifying Quadratic Curves
7.29. Singular Value Decomposition
 7.29.1. Constrained Optimization of Quadratic Forms 7.29.2. The Singular Values of a Matrix 7.29.3. Computing the Singular Values of a Matrix 7.29.4. Singular Value Decomposition of 2x2 Matrices 7.29.5. Singular Value Decomposition of 2x2 Matrices With Zero or Repeated Eigenvalues 7.29.6. Singular Value Decomposition of Larger Matrices 7.29.7. Singular Value Decomposition and the Pseudoinverse Matrix
8.
Applications of Linear Algebra
9 topics
8.30. Linear Least-Squares Problems
 8.30.1. The Least-Squares Solution of a Linear System (Without Collinearity) 8.30.2. The Least-Squares Solution of a Linear System (With Collinearity) 8.30.3. Finding a Least-Squares Solution Using QR Factorization 8.30.4. Weighted Least-Squares
8.31. Markov Chains
 8.31.1. Markov Chains 8.31.2. Steady-State Vectors
8.32. Linear Regression
 8.32.1. Linear Regression 8.32.2. Polynomial Regression 8.32.3. Multiple Linear Regression