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Linear Algebra | Incognition
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Linear Algebra
Vectors, matrices, and transformations — the geometric language machine learning is built on.
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75 lessons
Chapter 1: What Is a Vector?
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1
Direction and Magnitude
2
Vectors as Coordinates
3
Vector Addition & Subtraction
4
Scalar Multiplication
5
Vectors in AI: Representing Data
Chapter 2: Vector Operations & Similarity
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1
The Dot Product
2
Vector Length (Norm)
3
Unit Vectors & Normalization
4
Cosine Similarity
5
Vectors in AI: Word Embeddings & Similarity Search
Chapter 3: Introduction to Matrices
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What Is a Matrix?
2
Matrices as Real Data
3
Matrix Addition & Scalar Multiplication
4
The Transpose
5
Matrices in AI: Batches of Data
Chapter 4: Matrix Multiplication
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How Matrix Multiplication Works
2
Matrix-Vector Multiplication
3
Matrix-Matrix Multiplication
4
Why Order Matters (Non-Commutativity)
5
Matrix Multiplication in Neural Networks
Chapter 5: Linear Transformations
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What Is a Linear Transformation?
2
Where the Basis Vectors Land
3
A Gallery of Transformations
4
Composing Transformations
5
Linear Transformations in AI
Chapter 6: Systems of Linear Equations
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What Is a System of Linear Equations?
2
Systems as Matrix Equations
3
Solving by Elimination
4
How Many Solutions? One, None, or Infinite
5
Systems of Equations in AI
Chapter 7: Determinants & Inverses
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What Is a Determinant?
2
Calculating Determinants
3
What Is an Inverse Matrix?
4
Finding the Inverse
5
Determinants & Inverses in AI
Chapter 8: Vector Spaces, Span & Basis
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What Is a Vector Space?
2
Linear Combinations & Span
3
Linear Independence
4
Basis & Dimension
5
Vector Spaces in AI
Chapter 9: Eigenvalues & Eigenvectors
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What Is an Eigenvector?
2
What Is an Eigenvalue?
3
Finding Eigenvalues
4
Finding Eigenvectors
5
Eigenvalues & Eigenvectors in AI
Chapter 10: Orthogonality & Projections
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What Is Orthogonality?
2
Vector Projection
3
Orthogonal Bases
4
Orthogonal Matrices
5
Orthogonality & Projections in AI
Chapter 11: Matrix Decompositions
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Why Decompose a Matrix?
2
LU Decomposition
3
Eigendecomposition
4
QR Decomposition
5
Singular Value Decomposition in AI
Chapter 12: Norms & Distance in Machine Learning
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What Is a Norm?
2
The L1 and L∞ Norms
3
Unit Balls: Different Norms, Different Shapes
4
Distance Between Points
5
Norms in AI: Regularization & Sparsity
Chapter 13: Linear Algebra for Machine Learning
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The Design Matrix
2
Linear Regression as a Matrix Equation
3
The Normal Equations
4
When the Normal Equations Break
5
Linear Algebra in AI: Putting It All Together
Chapter 14: Linear Algebra for Deep Learning
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Tensors: Beyond Matrices
2
Stacking Layers Is Still Linear
3
Why Activation Functions Break Linearity
4
The Jacobian: Derivatives as Matrices
5
Linear Algebra in AI: Backpropagation
Chapter 15: Dimensionality Reduction & Embeddings
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The Curse of Dimensionality
2
PCA: Finding the Directions of Most Variance
3
How Many Dimensions Do We Need?
4
Embeddings: Compressing Meaning Into Vectors
5
Dimensionality Reduction in AI: The Full Pipeline
