# Linear Algebra

Linear algebra is the study of **linear maps** and the spaces
they act on. Vectors are the objects; a matrix is the map — and almost
every question in the subject is really a question about what that map
does to space.


_Figure 001 — A matrix as a transformation: the square grid shears into parallelograms,
carrying the basis vectors to its columns.
_

Read a matrix as a _verb_, not a table. Its columns record exactly
where the basis vectors land, and that alone fixes the fate of every other
vector — the grid deforms with them.


The first hard question is when a system _Ax = b_ has a solution at
all. Gaussian elimination answers it mechanically: clear each column below
its pivot and the matrix steps into row-echelon form, where the solutions
can simply be read off.


_Figure 002 — Gaussian elimination: entries below each pivot fall to zero, stepping the
matrix to row-echelon form.
_

Beneath the mechanics sit the **vector spaces** that organize
the answers. Independence, span, and basis measure how many directions a
set of vectors really reaches, and dimension counts them.


_Figure 003 — Span: every scalar combination of two independent vectors sweeps out the
whole plane they generate.
_

Some directions survive a transformation untouched in orientation. An
**eigenvector** is carried straight back onto its own line,
merely stretched by an eigenvalue — the axes along which the map acts most
simply.


_Figure 004 — Eigenvectors: a generic vector is rotated, but these special directions are
only scaled by <em>λ</em>.
_

Orthogonality turns geometry into computation. Projecting a vector onto a
subspace drops a perpendicular to the closest point in it — the idea behind
least squares, orthonormal bases, and the singular value decomposition.


_Figure 005 — Orthogonal projection: the foot p is the closest point on the line to b,
met at a right angle.
_

Determinants measure how a map scales volume and whether it is invertible;
an eigenbasis diagonalizes it; and the SVD factors _any_ matrix into
a rotation, a set of stretches, and another rotation.


The payoff is a single language for systems of equations, geometry, and
data — and a set of factorizations that make each of them computable,
even in floating point.


---

## Contents

### 1. Linear Equations in Linear Algebra

1. [Systems of Linear Equations and Row Reduction](/linear-algebra/linear-systems/systems-and-echelon-forms)
2. [Vector Equations and the Matrix Equation Ax = b](/linear-algebra/linear-systems/vector-and-matrix-equations)
3. [Solution Sets and Applied Linear Systems](/linear-algebra/linear-systems/solution-sets-and-applications)
4. [Linear Independence](/linear-algebra/linear-systems/linear-independence)
5. [Linear Transformations and Their Matrices](/linear-algebra/linear-systems/linear-transformations)

### 2. Matrix Algebra

1. [Matrix Operations](/linear-algebra/matrix-algebra/matrix-operations)
2. [The Inverse and the Invertible Matrix Theorem](/linear-algebra/matrix-algebra/matrix-inverse-and-invertibility)
3. [Block Matrices and the LU Factorization](/linear-algebra/matrix-algebra/partitioned-matrices-and-lu)
4. [Subspaces of Rⁿ, Dimension, and Rank](/linear-algebra/matrix-algebra/subspaces-dimension-rank)
5. [Applications: Leontief Economics and Computer Graphics](/linear-algebra/matrix-algebra/applications-leontief-and-graphics)

### 3. Determinants

1. [Introduction to Determinants](/linear-algebra/determinants/determinants-and-cofactors)
2. [Properties of Determinants](/linear-algebra/determinants/properties-of-determinants)
3. [Cramer's Rule, Volume, and Linear Transformations](/linear-algebra/determinants/cramer-volume-and-area)

### 4. Vector Spaces

1. [Vector Spaces and Subspaces](/linear-algebra/vector-spaces/vector-spaces-and-subspaces)
2. [Null Spaces, Column Spaces, and Linear Transformations](/linear-algebra/vector-spaces/null-and-column-spaces)
3. [Linearly Independent Sets and Bases](/linear-algebra/vector-spaces/bases-and-independent-sets)
4. [Coordinate Systems](/linear-algebra/vector-spaces/coordinate-systems)
5. [The Dimension of a Vector Space and Rank](/linear-algebra/vector-spaces/dimension-and-rank)
6. [Change of Basis](/linear-algebra/vector-spaces/change-of-basis)
7. [Applications: Difference Equations and Markov Chains](/linear-algebra/vector-spaces/difference-equations-and-markov)

### 5. Eigenvalues and Eigenvectors

1. [Eigenvectors and Eigenvalues](/linear-algebra/eigenvalues/eigenvectors-and-eigenvalues)
2. [The Characteristic Equation](/linear-algebra/eigenvalues/the-characteristic-equation)
3. [Diagonalization](/linear-algebra/eigenvalues/diagonalization)
4. [Eigenvectors and Linear Transformations](/linear-algebra/eigenvalues/eigenvectors-and-linear-transformations)
5. [Complex Eigenvalues](/linear-algebra/eigenvalues/complex-eigenvalues)
6. [Discrete and Continuous Dynamical Systems](/linear-algebra/eigenvalues/dynamical-systems)
7. [Iterative Estimates for Eigenvalues](/linear-algebra/eigenvalues/power-method)

### 6. Orthogonality and Least Squares

1. [Inner Product, Length, and Orthogonality](/linear-algebra/orthogonality-least-squares/inner-product-length-orthogonality)
2. [Orthogonal Sets and Orthogonal Projections](/linear-algebra/orthogonality-least-squares/orthogonal-sets-and-projections)
3. [The Gram-Schmidt Process and QR Factorization](/linear-algebra/orthogonality-least-squares/gram-schmidt-and-qr)
4. [Least-Squares Problems](/linear-algebra/orthogonality-least-squares/least-squares-problems)
5. [Applications to Linear Models](/linear-algebra/orthogonality-least-squares/least-squares-applications)
6. [Inner Product Spaces](/linear-algebra/orthogonality-least-squares/inner-product-spaces)

### 7. Symmetric Matrices, Quadratic Forms, and the SVD

1. [Diagonalization of Symmetric Matrices](/linear-algebra/symmetric-quadratic-svd/diagonalizing-symmetric-matrices)
2. [Quadratic Forms](/linear-algebra/symmetric-quadratic-svd/quadratic-forms)
3. [Constrained Optimization](/linear-algebra/symmetric-quadratic-svd/constrained-optimization)
4. [The Singular Value Decomposition](/linear-algebra/symmetric-quadratic-svd/singular-value-decomposition)
5. [Applications: Image Processing and Statistics](/linear-algebra/symmetric-quadratic-svd/svd-applications-pca-imaging)

### 8. Numerical Linear Algebra

1. [Numerical Thinking and Matrix Computation](/linear-algebra/numerical-linear-algebra/numerical-thinking-and-matrix-computation)
2. [LU and Cholesky Factorization in Practice](/linear-algebra/numerical-linear-algebra/lu-and-cholesky)
3. [Conditioning and Floating-Point Arithmetic](/linear-algebra/numerical-linear-algebra/conditioning-and-floating-point)
4. [Numerical Stability and Backward Error Analysis](/linear-algebra/numerical-linear-algebra/stability-and-error-analysis)
5. [QR, Householder, and Numerical Least Squares](/linear-algebra/numerical-linear-algebra/qr-and-numerical-least-squares)
6. [Numerical Eigenvalue Problems and the SVD](/linear-algebra/numerical-linear-algebra/numerical-eigenvalues-and-svd)

### 9. Geometry of Vector Spaces

1. [Affine Combinations](/linear-algebra/geometry-of-vector-spaces/affine-combinations)
2. [Affine Independence and Barycentric Coordinates](/linear-algebra/geometry-of-vector-spaces/affine-independence-and-barycentric-coordinates)
3. [Convex Combinations and Convex Sets](/linear-algebra/geometry-of-vector-spaces/convex-combinations-and-convex-sets)
4. [Hyperplanes and Polytopes](/linear-algebra/geometry-of-vector-spaces/hyperplanes-and-polytopes)
5. [Curves and Surfaces](/linear-algebra/geometry-of-vector-spaces/curves-and-surfaces)