Linear Algebra

Checkpoints

• Determinant Basis Defines the determinant of a square matrix recursively via cofactor expansion, explains its geometric meaning as signed volume scaling, and establishes its connection to invertibility of linear maps.
• Dimension & Rank Basis Proves that all bases of a finite-dimensional vector space have the same size, defines dimension as this invariant, explores dimensions of common spaces, and connects rank to dimension.
• Gauss-Jordan Elimination Basis Explains how to systematically reduce any matrix to reduced row echelon form using three elementary row operations, and why this process is the backbone of solving linear systems and computing ranks.
• General Linear Group Basis Shows that invertible n×n matrices form a group under multiplication, introduces GL(n, F) and its coordinate-free version GL(V), and explores key substructures including the special linear group SL(n, F).
• Image & Rank Basis Defines the image of a linear map as the set of all reachable output vectors, proves it is a subspace of the codomain, identifies it with the column space of a matrix, and introduces rank as its dimension.
• Invertible Matrix Basis Defines invertible matrices as those whose linear transformation can be undone, characterizes invertibility through Gauss-Jordan elimination, and shows how to compute the inverse by augmenting with the identity.
• Kernel Basis Defines the kernel of a linear map as the set of all inputs that map to zero, proves it is a subspace, connects it to injectivity and the homogeneous linear system, and introduces nullity as its dimension.
• Linear Equations Basis Shows how a system of linear equations is compactly expressed as a matrix equation $Ax = b$, and uses Gauss-Jordan elimination on the augmented matrix to classify solutions as none, unique, or infinitely many.
• Linear Map & Matrix Multiplication Basis Defines linear maps as structure-preserving functions between vector spaces, shows how every linear map is represented by a matrix, and explains why matrix multiplication is exactly the composition of linear maps.
• Linear Span Basis Defines the span of a set of vectors as the collection of all their linear combinations, proves it is always the smallest subspace containing the set, and introduces spanning sets as the foundation for bases.
• Linear Subspace Basis Defines a linear subspace as a subset of a vector space that is closed under addition and scalar multiplication, gives the key characterization theorem, and identifies the important examples that appear throughout linear algebra.
• Linearly Dependent Basis Defines linear dependence and independence for a set of vectors, explains what it means geometrically for vectors to carry redundant information, and connects independence to the uniqueness of solutions of a homogeneous system.
• LU Decomposition Basis Explains how to factor a square matrix into a lower-triangular and an upper-triangular matrix using Gaussian elimination, and how this factorization makes solving linear systems with multiple right-hand sides dramatically faster.
• Matrix Basis Introduces matrices as rectangular arrays that represent collections of vectors, covers their basic operations, and shows how the set of all m×n matrices itself forms a vector space.
• Properties of Determinant Basis Develops the key properties of the determinant: Laplace expansion along any row or column, how row operations change det, the transpose identity, multiplicativity, the adjugate-based inverse formula, and Cramer's rule.
• Rank-Nullity Theorem Basis Proves the rank-nullity theorem — that the dimension of a linear map's domain equals its rank plus its nullity — and draws out the consequences for square matrices and invertibility.
• Row and Column Spaces Basis Defines the row space and column space of a matrix as spans of its rows and columns, shows how row reduction gives bases for both, proves that their dimensions always agree, and introduces rank as this common value.
• Structure of Solutions of Linear Equations Basis Classifies the solution set of Ax = b using rank: no solution when rank(A) ≠ rank([A|b]), a unique solution when both ranks equal the number of unknowns, and an affine subspace of solutions otherwise. This checkpoint ties Gauss-Jordan elimination to the rank-nullity theorem to give a complete picture.