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∞

Four Bases from One Row Reduction

Row reduction on a single matrix provides the information needed to construct bases for the column space, null space, row space, and left null space.

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Your summary note

1. Column space and null space bases
  Record that the $r$ pivot columns of the original $A$ form a basis for the column space and the $n-r$ special solutions form a basis for the null space.
2. Row space basis = nonzero rows of $R$
  State that the first $r$ rows of $R$ are a basis, and carry $A$ with rows $(1,2,3,1)$ , $(1,1,2,1)$ , $(1,2,3,1)$ to $R$ and the basis $\{(1,0,1,1),(0,1,1,0)\}$ .
3. Left null space from $[A \mid I] \to [R \mid E]$
  Write $EA=R$ , take the rows of $E$ matching the zero rows of $R$ , and finish the example with $E$ delivering $(-1,0,1)$ as the single basis vector.
4. Row operations keep the row space, change the column space
  Give the argument that rows of $R$ are combinations of rows of $A$ and the steps reverse, then record $(1,1,1)$ as a vector in $C(A)$ but not $C(R)$ .
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