Solving Linear Systems

01Row Picture, Column Picture & Matrix Form 02Row Picture vs Column Picture in 3D 03Singular vs Non-Singular Matrices 04Two Ways to Compute Ax Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

01Gaussian Elimination & Back Substitution 02Elimination Matrices 03Permutation Matrices and Inverses Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

015 Views of Matrix Multiplication 02Matrix Inverses and Singularity 03Gauss-Jordan Elimination for Inverses Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

01Product Inverse & Transpose Rules 02A = LU Factorization 03Gaussian Elimination: ⅓n³ Operation Count 04Permutation Matrices: P⁻¹ = Pᵀ Problem set0/10 Problem set 20/10 MIT problem set0/6 Practice∞

01Permutation Matrices and PA = LU 02Why RᵀR Is Always Symmetric 03Vector Spaces and Subspaces 04The Column Space of a Matrix Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

Vector Spaces & Subspaces

01Vector Spaces and Subspaces 02Column Space of a Matrix 03Null Space of a Matrix Problem set0/10 Problem set 20/10 Practice∞

01Rank, Pivots, and Free Variables 02Special Solutions of the Null Space 03Reading the Null Space from RREF Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

01Solvability of Ax = b 02Particular Solutions and the Null Space 03Rank and the Four Cases of Ax = b Problem set0/10 Problem set 20/10 MIT problem set0/8 Practice∞

01Linear Independence & the Null Space 02Basis and Dimension 03The Rank-Nullity Theorem Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

01Rank and the Four Fundamental Subspaces 02Four Bases from One Row Reduction 03Matrices as Vector Spaces Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

01The Dimension Formula for Subspaces 02Differential Equations as Linear Algebra 03Rank-One Matrices and Outer Products 04The Four Fundamental Subspaces Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

01The Incidence Matrix as Difference Operator 02The Incidence Matrix of a Graph 03The Equilibrium Equation AᵀCAx = f Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Orthogonality & Least Squares

01Orthogonal Subspaces 02Null Space as Orthogonal Complement 03The Normal Equations Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

01The Projection Matrix P = aaᵀ/aᵀa 02Projection onto Subspaces 03Projection and Least Squares Problem set0/10 Problem set 20/10 MIT problem set0/6 Practice∞

01Complementary Projections: P and I−P 02Least Squares and the Normal Equations 03Least Squares as Projection 04Invertibility of AᵀA Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

01Orthogonal Matrices 02Projections onto Orthonormal Bases 03The Gram-Schmidt Process 04QR Factorization from Gram-Schmidt Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

Determinants

01The Three Axioms of the Determinant 02Determinant Properties from Three Axioms 03Determinants via Elimination 04Multiplicative and Transpose Properties Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

01The Permutation Formula for Determinants 02Cofactor Expansion 03Periodic Determinants of Tridiagonal Matrices Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

01The Cofactor Formula for Matrix Inverses 02Cramer's Rule 03Determinants as Volume Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

Eigenvalues & Dynamics

01Eigenvalues and Eigenvectors 02Finding Eigenvalues and Eigenvectors 03Complex Eigenvalues and Defective Matrices Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

01Diagonalization: A = SΛS⁻¹ 02Matrix Powers and the Stability Theorem 03Defective Matrices and Multiplicity 04Eigenvalues and the Fibonacci Sequence Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

01Solving du/dt = Au with Eigenvalues 02Eigenvalue Stability in the Complex Plane 03The Matrix Exponential e^(At)04The Companion Matrix Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

01Why 1 Is Always an Eigenvalue of Markov Matrices 02Markov Chain Steady States via Eigenvalues 03Markov Matrices and Steady States 04Orthonormal Bases and Fourier Coefficients Problem set0/10 Problem set 20/10 MIT problem set0/7 Practice∞

Symmetric Matrices & the SVD

01The Spectral Theorem 02Why Symmetric Matrices Have Real Eigenvalues 03Sylvester's Law of Inertia 04Testing Positive Definiteness Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

01The Conjugate Transpose 02Inverse of the Fourier Matrix 03The FFT Factorization Problem set0/10 Problem set 20/10 Practice∞

01Positive Definite Matrices 02Positive Definiteness via Pivots 03The Hessian and Positive Definiteness 04Principal Axis Theorem and the Ellipsoid Problem set0/10 Problem set 20/10 MIT problem set0/6 Practice∞

01Closure Properties of Positive Definite Matrices 02Matrix Similarity and Invariant Eigenvalues 03Why Eigenvalues Don't Classify Similarity 04Jordan Canonical Form Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

01The Singular Value Decomposition 02Reducing SVD to A^T A 03Computing the SVD: Two Worked Examples 04SVD and the Four Fundamental Subspaces Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

Linear Transformations & Applications

01Defining Linear Transformations 02Basis, Coordinates, and Linear Maps 03How a Linear Map Becomes a Matrix Problem set0/10 Problem set 20/10 Practice∞

01Image Compression as a Change of Basis 02Choosing a Basis: JPEG, Fourier, and Wavelets 03Change of Basis and Image Compression 04Change of Basis and Diagonalization Problem set0/10 Problem set 20/10 Practice∞

01Matrix Inverses and the Four Subspaces 02One-Sided Inverses and Projections 03The Pseudo-Inverse 04The Pseudo-Inverse via the SVD Problem set0/10 Problem set 20/10 Practice∞

Rank and the Four Fundamental Subspaces

Every m × n matrix of rank r gives rise to four fundamental subspaces whose dimensions are determined by m, n, and r.

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Your summary note

1. Definitions of $C(A)$ , $N(A)$ , $C(A^T)$ , $N(A^T)$
  Define each of the four subspaces for an $m \times n$ matrix, record which of $\mathbb{R}^m$ or $\mathbb{R}^n$ each lives in, and transpose $A^Ty=0$ into $y^TA=0^T$ .
2. Dimensions $r$ , $n-r$ , $r$ and $m-r$
  Write the dimension table for the four subspaces, and record that the pair sitting in $\mathbb{R}^n$ sums to $n$ while the pair in $\mathbb{R}^m$ sums to $m$ .
3. Row space and column space share the dimension $r$
  State that both have dimension equal to the rank, and work the example $(1,1,2)$ , $(2,2,5)$ , $(3,3,8)$ where two identical rows force the columns to be dependent.
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