The one factorisation every matrix has
Matrix Decompositions: LU, QR, SVD & More › Singular Value Decomposition
A = LLᵀ, complete
Cholesky Decomposition: Algorithm & Uses › What Cholesky Decomposition Is
A diagonal entry taken as a square root
Cholesky Decomposition: Algorithm & Uses › The Algorithm
A negative value under the square root
Cholesky Decomposition: Algorithm & Uses › Cholesky as a Positive Definiteness Test
A = LU, complete
LU Decomposition: Factorization & Solving › What LU Decomposition Is
The multiplier stored, not discarded
LU Decomposition: Factorization & Solving › Construction from Gaussian Elimination
A row swap recorded in the permutation
LU Decomposition: Factorization & Solving › Partial Pivoting: PA = LU
A = QR, complete
QR Decomposition: Methods & Applications › What QR Decomposition Is
The projection removed from a column
QR Decomposition: Methods & Applications › QR via Gram-Schmidt
The leftover normalised to length one
QR Decomposition: Methods & Applications › QR and Gram-Schmidt: The Connection
One eigenvalue times one rank-one piece
Spectral Decomposition: QDQᵀ & PCA › The Outer Product Form
Q orthogonal, Λ diagonal, Q transposed
Spectral Decomposition: QDQᵀ & PCA › Computing the Spectral Decomposition
QᵀQ = I
Spectral Decomposition: QDQᵀ & PCA › Properties of the Factors
Singular values as roots of eigenvalues
SVD: Singular Value Decomposition › Singular Values
The three factors assembled
SVD: Singular Value Decomposition › Computing the SVD
A as a sum of rank-one pieces
SVD: Singular Value Decomposition › The Outer Product Form
ad, the first half of ad − bc
Determinants: Formula, Properties & Applications › The 2×2 Formula
The six signed products of a 3×3
Determinants: Formula, Properties & Applications › The 3×3 Formula
Expansion along a row, minors highlighted
Determinants: Formula, Properties & Applications › Expanding by Minors and Cofactors
A grid stretched, then flattened
Determinants: Formula, Properties & Applications › Area, Volume, and Orientation
One column swapped for the right-hand side
Applications of Determinants › Cramer's Rule
The cofactor array, before transposing
Applications of Determinants › The Inverse via the Adjugate
The cross product written as a determinant
Applications of Determinants › The Cross Product as a Determinant
The alternating sign board
Cofactors: Minors, Laplace Expansion & Adjugate › Cofactors and the Sign Pattern
One row expanded, term by term
Cofactors: Minors, Laplace Expansion & Adjugate › Laplace Expansion Along a Row
Expanding down column 1
Cofactors: Minors, Laplace Expansion & Adjugate › Laplace Expansion Along a Column
The unit square after the matrix has acted
Determinant Geometry: Area, Volume & Orientation › Signed Area in Two Dimensions
A reflection: area kept, orientation reversed
Determinant Geometry: Area, Volume & Orientation › Orientation
The plane flattened onto a line
Determinant Geometry: Area, Volume & Orientation › Linear Transformations as Geometric Mappings
Only the diagonal product survives
Determinant Properties: Row Operations & Multiplicative Rule › Triangular and Diagonal Matrices
Subtracting λ from the diagonal
Eigenvalues & Eigenvectors: Definition & Examples › Rewriting as a Homogeneous System
Two directions the matrix leaves in place
Eigenvalues & Eigenvectors: Definition & Examples › Geometric Meaning
The determinant expanded into a polynomial
Characteristic Equation: Polynomial & Eigenvalues › The Characteristic Polynomial
Roots extracted from the characteristic polynomial
Characteristic Equation: Polynomial & Eigenvalues › Computing the Characteristic Polynomial: 2×2
A repeated eigenvalue, λ = 1
Characteristic Equation: Polynomial & Eigenvalues › Algebraic Multiplicity
Solving the shifted system for one eigenvalue
Characteristic Equation: Polynomial & Eigenvalues › Finding Eigenvectors After Finding Eigenvalues
A rotation has no real eigenvector
Complex Eigenvalues: Conjugate Pairs & Rotation › When Complex Eigenvalues Appear
Every direction turned, none left fixed
Complex Eigenvalues: Conjugate Pairs & Rotation › The 2×2 Case in Detail
Spiralling in, then spiralling out
Complex Eigenvalues: Conjugate Pairs & Rotation › Dynamical Systems Interpretation
Eigenvectors into P, eigenvalues into D
Matrix Diagonalization: PDP⁻¹ & Applications › Constructing the Diagonalization
A power collapsing to a diagonal power
Matrix Diagonalization: PDP⁻¹ & Applications › Matrix Powers
A Markov matrix after eight steps
Matrix Diagonalization: PDP⁻¹ & Applications › Recurrence Relations
Too few independent eigenvectors to fill P
Matrix Diagonalization: PDP⁻¹ & Applications › When Diagonalization Fails
A repeated eigenvalue with two directions, then with one
Eigenvalue Properties: Trace, Determinant & More › Algebraic and Geometric Multiplicity
Distinct eigenvalues, independent directions
Eigenvalue Properties: Trace, Determinant & More › Independence of Eigenvectors
Six steps of power iteration
Eigenvalue Properties: Trace, Determinant & More › The Dominant Eigenvalue and Power Iteration
The same product read down the columns, then across the rows
Linear Systems: Solving Ax = b › Writing a System in Matrix Form
[A | b] before any row operation
Linear Systems: Solving Ax = b › The Augmented Matrix
Unique, none, infinitely many
Linear Systems: Solving Ax = b › The Three Possible Outcomes
Row echelon form, then reduced
Linear Systems: Solving Ax = b › Row Echelon Form and Reduced Row Echelon Form
Echelon form: a staircase of pivots
Row Echelon Form (REF) & RREF › Row Echelon Form
Reduced form: pivots alone in their columns
Row Echelon Form (REF) & RREF › Reduced Row Echelon Form
A free column becomes a direction
Row Echelon Form (REF) & RREF › Pivot Columns and Free Columns
The contradiction row
Row Echelon Form (REF) & RREF › Detecting Inconsistency
Clearing one entry beneath a pivot
Gaussian Elimination: Row Reduction › Forward Elimination
Reduced row echelon form reached
Gaussian Elimination: Row Reduction › Gauss-Jordan Elimination
A row of zeros against a non-zero constant
Gaussian Elimination: Row Reduction › Worked Example: No Solution
A free column, and with it a family of solutions
Homogeneous Systems: Ax = 0 & Null Space › When Do Nontrivial Solutions Exist?
Special solutions, one per free column
Homogeneous Systems: Ax = 0 & Null Space › The Solution Set Is the Null Space
Ax = b is Ax = 0, shifted
Homogeneous Systems: Ax = 0 & Null Space › Homogeneous vs. Non-Homogeneous
b in the column space: consistent
Solvability: Existence & Uniqueness › The Existence Condition
Dependent columns: no uniqueness
Solvability: Existence & Uniqueness › The Uniqueness Condition
The three endings, side by side
Solvability: Existence & Uniqueness › The Three Cases Combined
The rank goes up when b is appended
Solvability: Existence & Uniqueness › The Rouché–Capelli Theorem
Opening scene, frozen
Cramer's Rule Visualizer › The Opening Scene: The System
Determinant of A, frozen
Cramer's Rule Visualizer › The Determinant of A
Second unknown, frozen
Cramer's Rule Visualizer › Replacing a Column
Singular system, frozen
Cramer's Rule Visualizer › When the Determinant Is Zero
Solution, frozen
Cramer's Rule Visualizer › The Solution
Augmented matrix, frozen
Linear System Solutions Visualizer › The Opening Scene: The Augmented Matrix
Forward pass, frozen
Linear System Solutions Visualizer › The Forward Pass
Backward pass, frozen
Linear System Solutions Visualizer › The Backward Pass
One solution, frozen
Linear System Solutions Visualizer › One Solution
No solution, frozen
Linear System Solutions Visualizer › No Solution
Infinitely many solutions, frozen
Linear System Solutions Visualizer › Infinitely Many Solutions
Inconsistent system, frozen
Least Squares Visualizer › No Exact Solution
Normal equations, frozen
Least Squares Visualizer › The Normal Equations
Solving for x̂, frozen
Least Squares Visualizer › Solving for x-hat
The residual, frozen
Least Squares Visualizer › The Residual Is Perpendicular
Projection matrix, frozen
Least Squares Visualizer › The Projection Matrix
Projection onto a line, frozen
Least Squares Visualizer › Projection onto a Line
Dependent columns, frozen
Least Squares Visualizer › When the Columns Are Dependent
Opening scene, frozen
Gaussian Elimination Calculator › Using the Calculator
Echelon form, frozen
Gaussian Elimination Calculator › The Two Target Forms
Reduced echelon form, frozen
Gaussian Elimination Calculator › The Two Target Forms
First pivot, frozen
Gaussian Elimination Calculator › Finding a Pivot
A row swap, frozen
Gaussian Elimination Calculator › Swapping Rows
A row scaling, frozen
Gaussian Elimination Calculator › Scaling to a Leading 1
An elimination below the pivot, frozen
Gaussian Elimination Calculator › Eliminating Below the Pivot
An elimination above the pivot, frozen
Gaussian Elimination Calculator › Clearing Above the Pivot
A skipped column, frozen
Gaussian Elimination Calculator › Columns With No Pivot
An inconsistent system, frozen
Gaussian Elimination Calculator › When There Is No Solution
Rows, columns, diagonal
Matrices: Definition, Types & Operations › Dimensions, Rows, and Columns
Ax built from columns
Matrices: Definition, Types & Operations › Matrices as Collections of Vectors
The unit grid after the matrix has acted
Matrices: Definition, Types & Operations › Matrices as Linear Transformations
The 2 × 2 recipe
Inverse Matrix: Formula & Computation › The 2×2 Inverse Formula
The identity reached on the left, the inverse on the right
Inverse Matrix: Formula & Computation › Computing the Inverse by Row Reduction
Cofactors assembled before transposing and dividing
Inverse Matrix: Formula & Computation › Computing the Inverse via the Adjugate
A + B, every entry settled
Matrix Operations: Rules & Properties › Matrix Addition
kA, entry by entry
Matrix Operations: Rules & Properties › Scalar Multiplication
αA + βB, the final sweep
Matrix Operations: Rules & Properties › Linear Combinations of Matrices
One entry of AB, from a row and a column
Matrix Operations: Rules & Properties › Matrix Multiplication — Definition
The product assembled from the rows
Matrix Operations: Rules & Properties › Matrix Multiplication — Column and Row Interpretations
Reflection across the main diagonal
Matrix Operations: Rules & Properties › The Transpose
A⁴ one product at a time
Matrix Operations: Rules & Properties › Matrix Powers
Elimination finished, pivots counted
Matrix Rank: Definition & Computation › Computing Rank via Row Reduction
rank 1 + nullity 1 = 2
Matrix Rank: Definition & Computation › The Rank-Nullity Theorem
The reduced form all four spaces are read from
Matrix Rank: Definition & Computation › Rank and the Four Fundamental Subspaces
Sweeping the main diagonal
Trace of a Matrix: Properties & Identities › Definition
Two products, one trace
Trace of a Matrix: Properties & Identities › The Cyclic Property
The identity, generated at this size
Types of Matrices: Properties & Examples › The Identity Matrix
Diagonal: only the diagonal is free
Types of Matrices: Properties & Examples › Diagonal Matrices
Upper triangular: everything below the diagonal is zero
Types of Matrices: Properties & Examples › Triangular Matrices
Symmetric: entries mirrored across the diagonal
Types of Matrices: Properties & Examples › Symmetric Matrices
Skew-symmetric: mirrored entries, opposite signs
Types of Matrices: Properties & Examples › Skew-Symmetric Matrices
Scene 0, frozen
Matrix Trace Visualizer › Scene 0: the Question, Before Anything Is Highlighted
Scene 1, frozen
Matrix Trace Visualizer › Scene 1: Revealing the Main Diagonal
Sweep step 2 of 4, frozen
Matrix Trace Visualizer › The Sweep: One Diagonal Entry at a Time
Final scene, frozen
Matrix Trace Visualizer › The Completed Trace
Cell-by-cell, step 6 of 12
Matrix Transpose Visualizer › Cell-by-Cell: the Definition, One Entry at a Time
Row-as-column, step 2 of 3
Matrix Transpose Visualizer › Row-as-Column: Moving a Whole Row at Once
Column-as-row, step 2 of 4
Matrix Transpose Visualizer › Column-as-Row: the Same Operation From the Other Side
Diagonal reflection, the geometric view
Matrix Transpose Visualizer › Diagonal Reflection: One Geometric Move
Opening scene, frozen
Matrix Addition and Subtraction Visualizer › The Opening Scene: Two Matrices of the Same Shape
Mid-sweep, frozen
Matrix Addition and Subtraction Visualizer › One Cell at a Time
Completed sum, frozen
Matrix Addition and Subtraction Visualizer › The Completed Sum
Subtraction, same point in the sweep
Matrix Addition and Subtraction Visualizer › Switching to Subtraction
Opening scene, frozen
Matrix Scalar Multiplication Visualizer › The Opening Scene: One Number and One Matrix
Mid-sweep, frozen
Matrix Scalar Multiplication Visualizer › One Cell at a Time
Completed product, frozen
Matrix Scalar Multiplication Visualizer › The Completed Product
Opening scene, frozen
Linear Combination of Matrices Visualizer › The Opening Scene: Two Matrices and Two Scalars
Phase 1, mid-sweep
Linear Combination of Matrices Visualizer › Phase 1: Scaling A by α
Phase 2, mid-sweep
Linear Combination of Matrices Visualizer › Phase 2: Scaling B by β
Phase 3, mid-sweep
Linear Combination of Matrices Visualizer › Phase 3: Adding the Two Scaled Matrices
Opening scene, frozen
Hadamard Product Visualizer › The Opening Scene: Two Matrices, Same Shape
Mid-sweep, frozen
Hadamard Product Visualizer › One Cell at a Time
Completed product, frozen
Hadamard Product Visualizer › The Completed Element-wise Product
Diagonal product, main diagonal revealed
Determinant Visualizer › Diagonal Product: the 2 × 2 Case
Sarrus's rule, positive diagonal 2 of 3
Determinant Visualizer › Sarrus's Rule: the 3 × 3 Shortcut
Cofactor along row 1, term 2 of 3
Determinant Visualizer › Cofactor Expansion Along a Row
Cofactor along column 1, term 2 of 3
Determinant Visualizer › Cofactor Expansion Along a Column
The sign pattern at n = 4
Determinant Visualizer › The Sign Pattern
Opening scene, frozen
Matrix Inverse Visualizer › The Opening Scene: A Matrix and an Empty Cofactor Grid
Phase 1, the centre cofactor
Matrix Inverse Visualizer › Phase 1: Cofactors, One Entry at a Time
Phase 2, the transpose
Matrix Inverse Visualizer › Phase 2: Transposing into the Adjugate
Phase 3, the determinant
Matrix Inverse Visualizer › Phase 3: The Determinant from the First Row
Phase 4, mid-sweep
Matrix Inverse Visualizer › Phase 4: Dividing by the Determinant
Opening scene, frozen
Matrix Rank Visualizer › The Opening Scene: A on the Left, a Copy on the Right
First pivot, frozen
Matrix Rank Visualizer › Finding a Pivot
First elimination, frozen
Matrix Rank Visualizer › Eliminating Below the Pivot
Skipped column, frozen
Matrix Rank Visualizer › Swapping Rows and Skipping Columns
Completed reduction, frozen
Matrix Rank Visualizer › The Completed Reduction
Opening scene, frozen
QR Decomposition Visualizer › The Opening Scene: A Equals Q Times R
Last coefficient, frozen
QR Decomposition Visualizer › Recording a Coefficient
Last subtraction, frozen
QR Decomposition Visualizer › Subtracting the Projections
Last normalization, frozen
QR Decomposition Visualizer › Normalizing into Q and the Diagonal of R
Completed factorization, frozen
QR Decomposition Visualizer › The Completed Factorization
Opening scene, frozen
LU Decomposition Visualizer › The Opening Scene: L Half Known, U a Copy of A
First pivot, frozen
LU Decomposition Visualizer › Marking a Pivot
Last elimination, frozen
LU Decomposition Visualizer › Eliminating and Recording the Multiplier
Forced row swap, frozen
LU Decomposition Visualizer › Swapping Rows: PA = LU
Completed factorization, frozen
LU Decomposition Visualizer › The Completed Factorization
The definition, frozen
Matrix Power Visualizer › The Definition: n Copies of A
First collapse, frozen
Matrix Power Visualizer › The First Collapse: A Squared
Second collapse, frozen
Matrix Power Visualizer › The Second Collapse: A Cubed and the Σ Summary
The result, frozen
Matrix Power Visualizer › The Result: A to the Fourth
Opening scene, frozen
Cholesky Decomposition Visualizer › The Opening Scene: L and Its Transpose
Second diagonal entry, frozen
Cholesky Decomposition Visualizer › A Diagonal Entry
Last off-diagonal entry, frozen
Cholesky Decomposition Visualizer › An Entry Below the Diagonal
Not positive definite, frozen
Cholesky Decomposition Visualizer › When the Matrix Is Not Positive Definite
Completed factorization, frozen
Cholesky Decomposition Visualizer › The Completed Factorization
A − λI, frozen
Eigenvalues and Eigenvectors Visualizer › Forming A Minus Lambda I
Characteristic polynomial, frozen
Eigenvalues and Eigenvectors Visualizer › The Characteristic Polynomial
Eigenvalues, frozen
Eigenvalues and Eigenvectors Visualizer › Finding the Roots
Eigenvector for λ = 11, frozen
Eigenvalues and Eigenvectors Visualizer › Reading an Eigenvector
Repeated eigenvalue, frozen
Eigenvalues and Eigenvectors Visualizer › A Repeated Eigenvalue with a Plane of Eigenvectors
Complex eigenvalues, frozen
Eigenvalues and Eigenvectors Visualizer › Complex Eigenvalues
Eigenvectors, frozen
Matrix Diagonalization Visualizer › The Eigenvector Matrix
P and D, frozen
Matrix Diagonalization Visualizer › Assembling P and D
P⁻¹, frozen
Matrix Diagonalization Visualizer › Inverting P
A = P D P⁻¹, frozen
Matrix Diagonalization Visualizer › The Factorization
A³ = P D³ P⁻¹, frozen
Matrix Diagonalization Visualizer › Powers Through the Diagonal
Markov chain, eighth power, frozen
Matrix Diagonalization Visualizer › A Markov Chain Settling Down
Defective, frozen
Matrix Diagonalization Visualizer › When It Fails
Starting point, frozen
Power Iteration Visualizer › The Starting Point
First step, frozen
Power Iteration Visualizer › The First Step
Third step, frozen
Power Iteration Visualizer › Closing In
After six steps, frozen
Power Iteration Visualizer › Converged
Slow convergence, frozen
Power Iteration Visualizer › When Convergence Is Slow
No dominant eigenvalue, frozen
Power Iteration Visualizer › When There Is No Dominant Eigenvalue
Eigenvectors to Q, frozen
Spectral Decomposition Visualizer › From Eigenvectors to Q
QᵀQ = I, frozen
Spectral Decomposition Visualizer › Q Transpose Q Equals I
A = Q Λ Qᵀ, frozen
Spectral Decomposition Visualizer › The Factorization
Rank-one expansion, frozen
Spectral Decomposition Visualizer › The Rank-One Expansion
Gram–Schmidt in the λ = 1 plane, frozen
Spectral Decomposition Visualizer › Gram–Schmidt Inside a Repeated Eigenspace
A single term, frozen
Spectral Decomposition Visualizer › A Single Rank-One Term
Not symmetric, frozen
Spectral Decomposition Visualizer › When the Matrix Is Not Symmetric
AᵀA, frozen
Singular Value Decomposition Visualizer › Forming A Transpose A
Singular values, frozen
Singular Value Decomposition Visualizer › The Singular Values
u₁ = A v₁ / σ₁, frozen
Singular Value Decomposition Visualizer › From v to u
A = U Σ Vᵀ, frozen
Singular Value Decomposition Visualizer › The Factorization
Rank-one expansion, frozen
Singular Value Decomposition Visualizer › The Rank-One Expansion
Completing U, frozen
Singular Value Decomposition Visualizer › Completing U
A rank-one matrix, frozen
Singular Value Decomposition Visualizer › A Rank-One Matrix
Row reduction, frozen
Four Fundamental Subspaces Visualizer › Row Reduction First
Column space, frozen
Four Fundamental Subspaces Visualizer › The Column Space
Row space, frozen
Four Fundamental Subspaces Visualizer › The Row Space
Null space, frozen
Four Fundamental Subspaces Visualizer › The Null Space
Left null space, frozen
Four Fundamental Subspaces Visualizer › The Left Null Space
Two orthogonalities, frozen
Four Fundamental Subspaces Visualizer › The Two Orthogonalities
A trivial null space, frozen
Four Fundamental Subspaces Visualizer › When a Null Space Is Trivial
QᵀQ = I, frozen
Orthogonal Matrices Visualizer › The Test
det Q = 1, frozen
Orthogonal Matrices Visualizer › Determinant Plus or Minus One
Lengths and angles, frozen
Orthogonal Matrices Visualizer › Lengths and Angles Survive
Q Qᵀ = I, frozen
Orthogonal Matrices Visualizer › The Inverse Is the Transpose
A reflection classified, frozen
Orthogonal Matrices Visualizer › Reading Off a Reflection
A rotation of space classified, frozen
Orthogonal Matrices Visualizer › The Axis of a Rotation in Space
Perpendicular, not unit, frozen
Orthogonal Matrices Visualizer › Perpendicular but Not Unit
Rank of V, frozen
Span and Membership Visualizer › Sizing the Span
Membership test, frozen
Span and Membership Visualizer › The Membership Test
Coordinates, frozen
Span and Membership Visualizer › Reading the Coordinates
Not in the span, frozen
Span and Membership Visualizer › Not in the Span
A dependent set, frozen
Span and Membership Visualizer › A Dependent Spanning Set
Identity, frozen
Square Matrix Types Generator › Identity and Zero
Zero, frozen
Square Matrix Types Generator › Identity and Zero
Scalar, frozen
Square Matrix Types Generator › Scalar and Diagonal
Diagonal, frozen
Square Matrix Types Generator › Scalar and Diagonal
Upper triangular, frozen
Square Matrix Types Generator › Triangular Matrices
Lower triangular, frozen
Square Matrix Types Generator › Triangular Matrices
Symmetric, frozen
Square Matrix Types Generator › Symmetric and Skew-Symmetric
Skew-symmetric, frozen
Square Matrix Types Generator › Symmetric and Skew-Symmetric
Random, frozen
Square Matrix Types Generator › Random Matrices
Step 0, frozen
Matrix Multiplication by Columns › The empty bracket
Entry 1 complete, then entry 2, frozen
Matrix Multiplication by Columns › One product at a time
First regrouping step, frozen
Matrix Multiplication by Columns › Lifting out a column
Steps 10 and 11, frozen
Matrix Multiplication by Columns › The statement and the arithmetic
Step 0, frozen
Matrix Multiplication by Rows › The empty bracket
Entry 1 complete, then entry 2, frozen
Matrix Multiplication by Rows › One product at a time
First regrouping step, frozen
Matrix Multiplication by Rows › Lifting out a row
Steps 10 and 11, frozen
Matrix Multiplication by Rows › The statement and the arithmetic
Step 0, frozen
Matrix Multiplication by Rows and Columns › The empty bracket
Both tab, a positive and a negative entry, frozen
Matrix Multiplication by Rows and Columns › One row, one column, one entry
Columns tab, first piece, frozen
Matrix Multiplication by Rows and Columns › Building a column of AB
Columns tab, both columns assembled, frozen
Matrix Multiplication by Rows and Columns › The finished product by columns
Rows tab, first piece, frozen
Matrix Multiplication by Rows and Columns › Building a row of AB
Rows tab, both rows assembled, frozen
Matrix Multiplication by Rows and Columns › The finished product by rows
Two orthogonal pairings of whole subspaces
Orthogonality: Vectors, Projections & Bases › The Four Fundamental Subspaces Revisited
The overlap removed from the second vector
Gram-Schmidt Process & QR Decomposition › The Algorithm: Two Vectors
Scaling the survivor to unit length
Gram-Schmidt Process & QR Decomposition › Normalization
The same run, recorded as two factors
Gram-Schmidt Process & QR Decomposition › The QR Decomposition
Paired products collapsed to one number
Inner Product: Dot Product, Norms & Angles › The Dot Product
A vector dotted with itself, then rooted
Inner Product: Dot Product, Norms & Angles › Length
The system has no solution
Least Squares: Normal Equations & Regression › The Problem
The residual, perpendicular to the column space
Least Squares: Normal Equations & Regression › The Geometric Interpretation
The normal equations assembled
Least Squares: Normal Equations & Regression › The Normal Equations
A best-fit line through scattered points
Least Squares: Normal Equations & Regression › Worked Example: Fitting a Line
P b = p
Least Squares: Normal Equations & Regression › The Projection Matrix
A set with every pair at right angles
Orthogonal & Orthonormal Sets and Bases › Orthogonal Sets
Columns of length one, pairwise orthogonal
Orthogonal & Orthonormal Sets and Bases › Orthogonal Matrices
The coefficient applied to the direction
Projections: Formulas & Matrix Properties › Projection onto a Vector
What is left after the projection is removed
Projections: Formulas & Matrix Properties › The Orthogonal Decomposition
The whole plane collapsed onto a line
Projections: Formulas & Matrix Properties › The Projection Matrix
What survives the transformation
Linear Transformations: Definition & Examples › Geometry
The same point, two sets of coordinates
Change of Basis & Similarity › The Change-of-Basis Matrix
A basis in which the map acts along the axes
Change of Basis & Similarity › Diagonalization as a Change of Basis
Scaling the unit square by 1.5
Geometric Transformations: Matrices in R² & R³ › Scaling
Rotating the unit square by 30°
Geometric Transformations: Matrices in R² & R³ › Rotations in R²
Reflection in an axis, then in a diagonal
Geometric Transformations: Matrices in R² & R³ › Reflections in R²
The plane dropped onto a line
Geometric Transformations: Matrices in R² & R³ › Projections
Shearing the unit square
Geometric Transformations: Matrices in R² & R³ › Shears
Shear after rotation, rotation after shear
Geometric Transformations: Matrices in R² & R³ › Combining Transformations
Area scaled, then area destroyed
Geometric Transformations: Matrices in R² & R³ › Determinant as Geometric Signature
A map reaching the whole plane
Image & Kernel: Injectivity and Surjectivity › The Image
A line crushed to the origin
Image & Kernel: Injectivity and Surjectivity › The Kernel
The basis vectors, and where they land
Matrix Representation of Transformations › Constructing the Standard Matrix
Two maps applied in each order
Matrix Representation of Transformations › Composition Corresponds to Matrix Multiplication
A shear, then its inverse
Matrix Representation of Transformations › The Identity and the Inverse
Order irrelevant, then decisive
Linear Transformation Properties › Composition
Enough directions, then not enough
Vector Spaces: Axioms, Basis & Dimension › Independence and Span
Two independent vectors spanning R²
Basis of a Vector Space and Coordinates › Basis: Definition
The same point, read against a skewed basis
Basis of a Vector Space and Coordinates › Coordinates
Dimension two, then dimension one
Dimension: Basis Size & Invariant › Definition
Pivot columns above, free-column solutions below
Dimension: Basis Size & Invariant › The Rank-Nullity Theorem as a Dimension Statement
Pivot columns marked as a basis for the column space
Four Fundamental Subspaces of a Matrix › The Column Space
The row space from R
Four Fundamental Subspaces of a Matrix › The Row Space
One special solution per free column
Four Fundamental Subspaces of a Matrix › The Null Space
The left null space from Aᵀ
Four Fundamental Subspaces of a Matrix › The Left Null Space
The two orthogonality pairings, side by side
Four Fundamental Subspaces of a Matrix › Orthogonal Complements
Independent, then very nearly dependent
Linear Independence: Definition & Tests › Geometric Interpretation in Rⁿ
A dependence relation found among the columns
Linear Independence: Definition & Tests › Testing Independence: The Homogeneous System
A span filling the plane, then collapsing to a line
Span: Linear Combinations & Spanning Sets › Geometric Interpretation
Solving for the weights that reach b
Span: Linear Combinations & Spanning Sets › Testing Whether a Vector Is in a Span
Two vectors cannot span R³
Span: Linear Combinations & Spanning Sets › Testing Whether a Set Spans Rⁿ
A null space that is just {0}
Subspaces: Definition, Test & Examples › Trivial Subspaces
The two non-trivial subspaces of R²
Subspaces: Definition, Test & Examples › Subspaces of R² and R³
The null space read off the reduced form
Subspaces: Definition, Test & Examples › The Null Space
One vector, three places
Vectors: Definition, Operations & Products › Types of Vectors
Addition finished, component by component
Vectors: Definition, Operations & Products › Basic Vector Operations
u + v, every component settled
Vector Operations: Addition, Subtraction & Scaling › Vector Addition
Commutativity, drawn
Vector Operations: Addition, Subtraction & Scaling › Properties of Addition
Subtraction as the arrow between two tips
Vector Operations: Addition, Subtraction & Scaling › Vector Subtraction
cv, every component scaled
Vector Operations: Addition, Subtraction & Scaling › Scalar Multiplication
The symbolic determinant laid out
Cross Product: Formula, Right-Hand Rule & Properties › Algebraic Definition
Length as area
Cross Product: Formula, Right-Hand Rule & Properties › Geometric Interpretation
Order decides direction
Cross Product: Formula, Right-Hand Rule & Properties › Direction and the Right-Hand Rule
Products formed, then summed to one number
Dot Product: Formula, Angle & Projection › Algebraic Definition
Where the cosine comes from
Dot Product: Formula, Angle & Projection › Geometric Definition
Positive, zero, negative
Dot Product: Formula, Angle & Projection › Sign of the Dot Product
The coefficient applied to the direction vector
Dot Product: Formula, Angle & Projection › Orthogonal Projection
αu + βv, third phase frozen
Linear Combinations: Span & Spanning Sets › Definition
Two vectors spanning a plane, then collapsing to a line
Linear Combinations: Span & Spanning Sets › Geometric Interpretation
Av assembled from the columns of A
Linear Combinations: Span & Spanning Sets › Linear Combinations and Systems of Equations
Squares summed, root about to be taken
Vector Magnitude: Norm, Distance & Unit Vectors › Magnitude in Two and Three Dimensions
Unit vectors in every direction
Vector Magnitude: Norm, Distance & Unit Vectors › Unit Vectors
Dividing through by the norm
Vector Magnitude: Norm, Distance & Unit Vectors › Normalization
One vector a multiple of the other
Vector Properties: Magnitude, Direction & Orthogonality › Parallelism
Opening scene, frozen
Vector Addition and Subtraction Visualizer › The Opening Scene: Two Vectors of the Same Length
Mid-sweep, frozen
Vector Addition and Subtraction Visualizer › One Component at a Time
Completed sum, frozen
Vector Addition and Subtraction Visualizer › The Completed Sum
Subtraction, same point in the sweep
Vector Addition and Subtraction Visualizer › Switching to Subtraction
Opening scene, frozen
Vector Scalar Multiplication Visualizer › The Opening Scene: One Number and One Vector
Mid-sweep, frozen
Vector Scalar Multiplication Visualizer › One Component at a Time
Completed product, frozen
Vector Scalar Multiplication Visualizer › The Completed Product
Opening scene, frozen
Linear Combination of Vectors Visualizer › The Opening Scene: Two Vectors and Two Scalars
Phase 1, mid-sweep
Linear Combination of Vectors Visualizer › Phase 1: Scaling u by α
Phase 2, mid-sweep
Linear Combination of Vectors Visualizer › Phase 2: Scaling v by β
Phase 3, mid-sweep
Linear Combination of Vectors Visualizer › Phase 3: Adding the Two Scaled Vectors
Opening scene, frozen
Inner Product Visualizer › The Opening Scene: Two Vectors, One Number
Mid-sweep, frozen
Inner Product Visualizer › Pairing, Multiplying, Accumulating
Completed inner product, frozen
Inner Product Visualizer › The Completed Inner Product
Opening scene, frozen
Cross Product of Vectors Visualizer › The Opening Scene: Two Vectors in Three Dimensions
Component method, middle step
Cross Product of Vectors Visualizer › The Component Sweep
Determinant method, the j step
Cross Product of Vectors Visualizer › The Determinant Expansion
Completed cross product, frozen
Cross Product of Vectors Visualizer › The Completed Product
Opening scene, frozen
Vector Magnitude and Unit Vector Visualizer › The Opening Scene: One Vector, One Number
Phase 1, mid-sweep
Vector Magnitude and Unit Vector Visualizer › Phase 1: Squaring Each Component
Phase 2, the sum and root
Vector Magnitude and Unit Vector Visualizer › Phase 2: Summing and Taking the Root
Phase 3, mid-sweep
Vector Magnitude and Unit Vector Visualizer › Phase 3: Normalizing to a Unit Vector
Opening scene, frozen
Vector Projection Visualizer › The Opening Scene: Two Vectors and a Dot Product to Come
Phase 1, mid-sweep
Vector Projection Visualizer › Phase 1: Pairing Components Toward the Dot Product
Phase 4, the coefficient
Vector Projection Visualizer › Phase 4: The Coefficient
Phase 5, mid-sweep
Vector Projection Visualizer › Phase 5: Scaling v by the Coefficient
Phase 6, mid-sweep
Vector Projection Visualizer › Phase 6: The Perpendicular Remainder
Opening scene, frozen
Outer Product of Vectors Visualizer › The Opening Scene: A Column, a Row, and a Grid
Cell method, the centre entry
Outer Product of Vectors Visualizer › Cell by Cell: The Definition Made Literal
Row method, row 2
Outer Product of Vectors Visualizer › Row by Row: Each Row Is a Copy of v Transpose
Column method, column 2
Outer Product of Vectors Visualizer › Column by Column: Each Column Is a Copy of u
Completed outer product, frozen
Outer Product of Vectors Visualizer › The Completed Product
Opening scene, frozen
Gram-Schmidt Process Visualizer › The Opening Scene: The Input Set as Rows
First vector kept, frozen
Gram-Schmidt Process Visualizer › Keeping the First Vector
Last subtraction, frozen
Gram-Schmidt Process Visualizer › Subtracting the Projections
First normalization, frozen
Gram-Schmidt Process Visualizer › Normalizing
Completed orthonormal set, frozen
Gram-Schmidt Process Visualizer › The Completed Orthonormal Set
Identity, frozen at t = 1
Linear Transformation 2D Visualizer › The Identity: the Transformation That Does Nothing
Rotate 45°, frozen at t = 1
Linear Transformation 2D Visualizer › Full Rank: Reshaping Without Losing Anything
Project to the x-axis, frozen at t = 1
Linear Transformation 2D Visualizer › Rank 1: Collapsing the Plane Onto a Line
The zero map, frozen at t = 1
Linear Transformation 2D Visualizer › Rank 0: the Zero Map
Distinct real eigenvalues, frozen
Eigenvectors 2D Visualizer › Distinct Real Eigenvalues: Two Fixed Directions
Repeated eigenvalue, full eigenspace, frozen
Eigenvectors 2D Visualizer › Repeated Eigenvalue With a Full Set of Directions
Defective matrix, frozen
Eigenvectors 2D Visualizer › Defective: a Repeated Eigenvalue Short of Directions
Complex eigenvalues, frozen
Eigenvectors 2D Visualizer › Complex Eigenvalues: No Real Direction Survives
Full rank (rotation), frozen
Kernel and Image 2D Visualizer › Full Rank: Nothing Collapses
Rank 1 (projection to x), frozen
Kernel and Image 2D Visualizer › Rank 1: a Kernel Line and an Image Line
Rank 0 (the zero map), frozen
Kernel and Image 2D Visualizer › Rank 0: Everything Goes to the Origin
Rotated 30° basis, frozen
Change of Basis 2D Visualizer › Natural Bases: Rotated and Stretched Axes
Skewed basis, frozen
Change of Basis 2D Visualizer › Non-Orthogonal Bases: Parallelograms Instead of Squares
Y-flipped basis, frozen
Change of Basis 2D Visualizer › Orientation Reversed: a Negative Determinant
Collinear pair, frozen
Change of Basis 2D Visualizer › Degenerate: When the Pair Is Not a Basis
Orthogonal pair, frozen
Span and Linear Independence 2D Visualizer › Independent Pairs
Near-aligned pair, frozen
Span and Linear Independence 2D Visualizer › Near-Dependence
b = 2a, frozen
Span and Linear Independence 2D Visualizer › Dependent Pairs
a = 0, frozen
Span and Linear Independence 2D Visualizer › Edge Cases
x-axis projection, frozen
Projection 2D Visualizer › Projection onto a Coordinate Axis
y = x projection, frozen
Projection 2D Visualizer › Projection onto a Diagonal
30° projection, frozen
Projection 2D Visualizer › Projection onto a Custom Angle
v on the kernel, frozen
Projection 2D Visualizer › A Vector on the Kernel
x-axis reflection, frozen
Reflection 2D Visualizer › Reflection across a Coordinate Axis
y = x reflection, frozen
Reflection 2D Visualizer › Reflection across a Diagonal
30° reflection, frozen
Reflection 2D Visualizer › Reflection across a Custom Angle
Morph at t = 0.5, frozen
Reflection 2D Visualizer › Halfway through the Morph: The Projection
Two rotations, end of the AB run
Matrix Composition 2D Visualizer › A Commuting Pair: Two Rotations
Shear then rotate, end of the AB run
Matrix Composition 2D Visualizer › A Non-Commuting Pair: Shear and Rotation
Two reflections, end of the AB run
Matrix Composition 2D Visualizer › A Reveal: Two Reflections Make a Rotation
Shear and its inverse, end of the AB run
Matrix Composition 2D Visualizer › A Reveal: A Shear and Its Inverse
45° rotation, frozen
Complex Eigenvalues 2D Visualizer › A Plain Rotation
Skewed 60° rotation, frozen
Complex Eigenvalues 2D Visualizer › A Rotation in a Skewed Basis
The columns of P, frozen
Complex Eigenvalues 2D Visualizer › The Columns of P
Spiral in, frozen
Complex Eigenvalues 2D Visualizer › Spiralling In
Spiral out, frozen
Complex Eigenvalues 2D Visualizer › Spiralling Out
Skewed spiral in, frozen
Complex Eigenvalues 2D Visualizer › A Skewed Spiral
General A × B, the opening scene
Matrix Multiplication Visualizer › General A × B: Any Compatible Pair of Shapes
Matrix × vector, c1,1 at term 2 of 3
Matrix Multiplication Visualizer › Matrix × Vector: A Column on the Right
Vector × matrix, c1,1 at term 2 of 3
Matrix Multiplication Visualizer › Vector × Matrix: A Row on the Left
B × A with B 3×3 and A 2×3
Matrix Multiplication Visualizer › Order and the Undefined Product
Row · column, c1,2 at term 2 of 3
Matrix Multiplication Visualizer › Row · Column: One Cell at a Time
Column by column, column 2 of C at term 2 of 3
Matrix Multiplication Visualizer › Column by Column: Combining the Columns of A
Row by row, row 2 of C at term 2 of 3
Matrix Multiplication Visualizer › Row by Row: Combining the Rows of B
Sum of outer products, contribution 2 of 3
Matrix Multiplication Visualizer › Sum of Outer Products: All of C at Once