Unit · year 1
MU-103 · Linear Algebra I
Threads structure25 lectures6 theorems
Vectors, matrices, and linear maps — the language of everything linear.
PREREQUISITES
Lectures
| L01 | Systems of Linear Equations and Gaussian Elimination — |
| L02 | Row Echelon Form and Rank — |
| L03 | Matrices and Matrix Algebra — |
| L04 | The Inverse of a Matrix |
| L05 | Elementary Matrices and LU Factorisation — |
| L06 | Vector Spaces: Axioms and Examples — |
| L07 | Subspaces, Span, and Linear Independence — |
| L08 | Basis and Dimension — |
| L09 | Coordinates and Change of Basis — |
| L10 | Linear Transformations — |
| L11 | Kernel and Image |
| L12 | The Rank–Nullity Theorem |
| L13 | Matrix Representation of Linear Maps — |
| L14 | The Determinant: Definition and Properties — |
| L15 | Cofactor Expansion and Computation — |
| L16 | Multiplicativity of the Determinant |
| L17 | Cramer's Rule and the Adjugate |
| L18 | The Invertible Matrix Theorem |
| L19 | Inner Products and Norms — |
| L20 | The Cauchy–Schwarz Inequality |
| L21 | Orthogonality and Orthogonal Complements — |
| L22 | The Gram–Schmidt Process |
| L23 | Orthogonal Matrices and QR Factorisation |
| L24 | Eigenvalues and Eigenvectors: A First Look — |
| L25 | Synthesis: Solving, Structure, and Geometry |
Theorems in this unit
T-015
The rank–nullity theorem
For a linear map, rank plus nullity equals the dimension of the domain.
T-016
The invertible matrix theorem
A dozen conditions on a square matrix are all equivalent to invertibility.
T-017
Multiplicativity of the determinant
det(AB) = det(A)det(B).
T-018
The Cauchy–Schwarz inequality
The inner product of two vectors is bounded by the product of their norms.
T-019
The Gram–Schmidt process
Any basis can be turned into an orthonormal one.
T-020
Cramer's rule
Solutions of a square linear system as ratios of determinants.