▦ Linear Algebra
Vectors, matrices, determinants, eigenvalues, vector spaces and decompositions.
104 topics
- Scalars and Vectors (Intermediate) – A scalar is a number; a vector has magnitude and direction.
- Vector Notation and Components (Intermediate) – Write vectors as ordered lists of coordinates.
- Vector Addition (Intermediate) – Add component-wise; geometrically tip-to-tail.
- Scalar Multiplication (Intermediate) – Scale a vector’s length.
- Vector Magnitude (Intermediate) – Length of a vector.
- Unit Vectors (Intermediate) – Vectors of length 1, found by normalising.
- Dot Product (Advanced) – Sum of component products.
- Angle Between Vectors (Advanced) – Cosine from the dot product.
- Orthogonality (Advanced) – Vectors are orthogonal when their dot product is 0.
- Projection (Advanced) – Component of one vector along another.
- Cross Product (Advanced) – A vector perpendicular to two vectors in 3D.
- Triple Product (Expert) – Scalar triple product gives parallelepiped volume.
- Lines and Planes in Vector Form (Advanced) – Parametric and normal equations.
- Matrices: Definition (Intermediate) – A rectangular array of numbers with m rows and n columns.
- Matrix Addition and Scaling (Intermediate) – Entrywise operations on matrices of equal size.
- Matrix Multiplication (Advanced) – Row-by-column products; not commutative.
- Identity Matrix (Advanced) – The matrix I with 1s on the diagonal; AI = A.
- Transpose (Advanced) – Swap rows and columns.
- Special Matrices (Advanced) – Diagonal, triangular, symmetric, orthogonal.
- Linear Systems as Matrices (Advanced) – Write simultaneous equations as Ax = b.
- Augmented Matrix (Advanced) – Combine coefficients and constants for row operations.
- Row Operations (Advanced) – Swap rows, scale a row, add a multiple of one row to another.
- Gaussian Elimination (Advanced) – Reduce to row echelon form then back-substitute.
- Gauss–Jordan Elimination (Advanced) – Reduce to reduced row echelon form.
- Consistency and Solution Types (Advanced) – Unique, infinite or no solutions.
- Homogeneous Systems (Expert) – Ax = 0 always has the trivial solution.
- Matrix Inverse (Advanced) – A matrix A⁻¹ with AA⁻¹ = I.
- Inverse by Row Reduction (Advanced) – Row reduce [A | I] to [I | A⁻¹].
- Determinant of 2×2 (Advanced) – ad − bc; zero means not invertible.
- Determinant of 3×3 (Advanced) – Expand along a row or use the rule of Sarrus.
- Cofactor Expansion (Expert) – Compute determinants recursively.
- Properties of Determinants (Expert) – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants (Expert) – Area or volume scaling factor.
- Cramer’s Rule for Systems (Expert) – Solve systems using determinants.
- Adjugate Matrix (Expert) – Transpose of the cofactor matrix.
- Linear Transformations (Advanced) – Functions preserving addition and scalar multiplication.
- Matrix of a Transformation (Advanced) – Columns are images of basis vectors.
- Rotation and Reflection Matrices (Advanced) – Matrices that rotate or reflect the plane.
- Scaling and Shear Matrices (Advanced) – Matrices that stretch or slant space.
- Composition of Transformations (Advanced) – Multiply matrices to compose maps.
- Vector Spaces (Expert) – Sets closed under addition and scalar multiplication with eight axioms.
- Subspaces (Expert) – Subsets that are themselves vector spaces.
- Span (Expert) – All linear combinations of given vectors.
- Linear Independence (Expert) – No vector is a combination of the others.
- Basis (Expert) – A linearly independent spanning set.
- Dimension (Expert) – The number of vectors in a basis.
- Coordinates Relative to a Basis (Expert) – Express a vector with respect to a chosen basis.
- Change of Basis (Expert) – Convert coordinates between bases.
- Column Space and Row Space (Expert) – Spans of columns and rows of a matrix.
- Null Space (Expert) – All solutions of Ax = 0.
- Rank (Expert) – The dimension of the column space.
- Rank–Nullity Theorem (Expert) – Rank plus nullity equals number of columns.
- Four Fundamental Subspaces (Expert) – Column, row, null and left-null spaces.
- Inner Product Spaces (Expert) – Vector spaces with a notion of angle and length.
- Orthonormal Bases (Expert) – Mutually orthogonal unit vectors.
- Gram–Schmidt Process (Expert) – Turn any basis into an orthonormal one.
- QR Decomposition (Expert) – Factor A = QR with orthogonal Q and triangular R.
- Least Squares (Expert) – Best approximate solution minimising error.
- Linear Regression via Matrices (Expert) – Fit a line using normal equations.
- Orthogonal Projections (Expert) – Closest point in a subspace.
- Orthogonal Matrices (Expert) – Preserve length: QᵀQ = I.
- Eigenvalues and Eigenvectors (Expert) – Vectors only scaled by a transformation.
- Characteristic Polynomial (Expert) – Eigenvalues are roots of det(A − λI).
- Eigenspaces (Expert) – The set of eigenvectors for an eigenvalue plus zero.
- Diagonalization (Expert) – Write A = PDP⁻¹ with D diagonal.
- Powers of Matrices (Expert) – Use diagonalization to compute Aⁿ.
- Spectral Theorem (Expert) – Symmetric real matrices have orthogonal eigenvectors.
- Positive Definite Matrices (Expert) – xᵀAx > 0 for all x ≠ 0.
- Quadratic Forms (Expert) – Expressions xᵀAx classified by eigenvalues.
- Singular Value Decomposition (Expert) – Factor any matrix as UΣVᵀ.
- Principal Component Analysis (Expert) – Find directions of greatest variance using eigenvectors.
- Pseudoinverse (Expert) – Generalised inverse A⁺ for non-square matrices.
- Trace (Expert) – Sum of the diagonal entries; equals sum of eigenvalues.
- Cayley–Hamilton Theorem (Expert) – Every matrix satisfies its characteristic equation.
- Jordan Canonical Form (Expert) – A nearly diagonal form for any complex matrix.
- Matrix Exponential (Expert) – Defined by the series eᴬ.
- LU Decomposition (Expert) – Factor A into lower and upper triangular matrices.
- Cholesky Decomposition (Expert) – Factor a positive definite matrix as LLᵀ.
- Markov Chains and Matrices (Expert) – Transition matrices and steady states.
- Google PageRank (Expert) – Ranking pages via the dominant eigenvector.
- Computer Graphics Transforms (Advanced) – Matrices that move, rotate and scale 3D objects.
- Homogeneous Coordinates (Expert) – Represent translations with matrix multiplication.
- Linear Recurrences via Matrices (Expert) – Compute Fibonacci numbers using matrix powers.
- Fibonacci Matrix (Expert) – [[1,1],[1,0]]ⁿ gives Fibonacci numbers.
- Block Matrices (Expert) – Partition matrices to simplify computation.
- Kronecker Product (Expert) – A larger matrix from two smaller matrices.
- Hadamard Product (Expert) – Entrywise matrix multiplication.
- Norms of Vectors and Matrices (Expert) – Measures of size such as ℓ¹, ℓ², ℓ∞ and Frobenius.
- Condition Number (Expert) – Sensitivity of Ax = b to errors.
- Power Iteration (Expert) – Numerical method to find the dominant eigenvector.
- Gershgorin Circle Theorem (Expert) – Bounds on eigenvalue locations.
- Linear Maps Between Spaces (Expert) – Kernel, image and isomorphism.
- Dual Space (Expert) – Space of linear functionals on V.
- Quotient Spaces (Expert) – Vector space formed by identifying vectors modulo a subspace.
- Direct Sums (Expert) – Decompose spaces into independent subspaces.
- Tensors Introduction (Expert) – Multilinear maps generalising vectors and matrices.
- Linear Algebra over Finite Fields (Expert) – Matrices with entries modulo a prime; used in coding.
- Error-Correcting Codes (Expert) – Hamming and linear codes detect and repair errors.
- Network Flows and Matrices (Expert) – Incidence matrices of graphs.
- Linear Algebra in Machine Learning (Expert) – Embeddings, weights and gradient steps as matrix operations.
- Convolution and Circulant Matrices (Expert) – Matrices diagonalised by the Fourier transform.
- Discrete Fourier Transform (Expert) – Change of basis to frequency space.
- Affine Spaces (Expert) – Spaces of points with translation vectors.
- Eigen Methods for ODEs (Expert) – Solve x′ = Ax with eigen decomposition.