Linear Algebra Done Right

Vector Spaces

01Complex Numbers as Pairs of Real Numbers 02Lists and Fⁿ: Coordinate Addition and a First Proof 03Vectors in R², Scalar Multiplication, and Fields Problem set0/10 Problem set 20/10

01Definition of a Vector Space: Fⁿ, F^∞ and Function Spaces 02Unique Zero and Inverses, 0v = 0, and the Subspace Test 03Sums of Subspaces: The Smallest Subspace Containing Them 04Direct Sums: Only One Way to Write Zero Problem set0/10 Problem set 20/10

Finite-Dimensional Spaces

01Linear Combinations, Span, and Finite-Dimensional Spaces 02Linear Independence and the Linear Dependence Lemma 03Independent Lists Are No Longer Than Spanning Lists Problem set0/10 Problem set 20/10

01Bases: Reduce a Spanning List, Extend an Independent List 02Dimension: Why Every Basis Has the Same Length 03The Dimension of a Sum of Two Subspaces Problem set0/10 Problem set 20/10

Linear Maps

01Linear Maps and the Linear Map Lemma: Values on a Basis 02Sums, Scalar Multiples and Products of Linear Maps 03Null Space, Range, Injectivity and Surjectivity 04The Fundamental Theorem of Linear Maps and Linear Systems Problem set0/10 Problem set 20/10

01The Matrix of a Linear Map: Each Column Is Tvₖ 02Matrix Operations and Why Multiplication Is Row Times Column 03Rows, Columns and Linear Combinations in Matrix Products 04Column Rank Equals Row Rank: Column-Row Factorization Problem set0/10 Problem set 20/10

01Invertible Linear Maps: Injective and Surjective 02Equal Dimensions: Injective or Surjective Means Invertible 03Isomorphic Spaces and the Dimension of L(V, W)04The Matrix of a Vector: M(Tv) = M(T)M(v)05Change of Basis for an Operator: A = C⁻¹BC Problem set0/10 Problem set 20/10

01Products of Vector Spaces: When Dimensions Add 02Quotient Spaces V/U: Parallel Lines as Vectors 03The Quotient Map and V/null T ≅ range T Problem set0/10 Problem set 20/10

01The Dual Basis Reads Coefficients and T′ Reverses Arrows 02The Annihilator U⁰ and dim U⁰ = dim V − dim U 03Null T′ and Range T′ as Annihilators; M(T′) Is the Transpose Problem set0/10 Problem set 20/10

Polynomials and Eigenvalues

01Conjugates, |z| and the Complex Triangle Inequality 02Zeros, Factors and the Division Algorithm via a Basis 03Fundamental Theorem of Algebra and Complete Factorization 04Real Polynomials: Linear and Irreducible Quadratic Factors Problem set0/10 Problem set 20/10

01Invariant Subspaces and Eigenvalues: Tv = λv 02Eigenvectors, T − λI and At Most dim V Eigenvalues 03p(T) and Why Every Complex Operator Has an Eigenvalue 04The Minimal Polynomial: Existence, Uniqueness and Computation 05Eigenvalues Are the Zeros of the Minimal Polynomial 06Why Odd Dimension Forces a Real Eigenvalue Problem set0/10 Problem set 20/10

01Upper-Triangular Matrices and Invariant Spans 02Eigenvalues Sit on the Upper-Triangular Diagonal 03The Minimal Polynomial Decides Upper-Triangularity Problem set0/10 Problem set 20/10

01Diagonalizable Operators and Direct Sums of Eigenspaces 02Four Equivalent Tests for Diagonalizability 03Distinct Linear Factors Decide Diagonalizability 04The Gershgorin Disk Theorem: Where Eigenvalues Live 05Commuting Operators Share a Diagonalizing Basis 06Commuting Operators Over C: One Upper-Triangular Basis Problem set0/10 Problem set 20/10

Inner Product Spaces

01Inner Products: Why Complex Spaces Need a Conjugate 02Norms, Orthogonality and the Pythagorean Theorem 03Cauchy-Schwarz and the Triangle Inequality Problem set0/10 Problem set 20/10

01Orthonormal Bases: Bessel's Inequality and Parseval 02Gram-Schmidt, Orthonormal Bases and Schur's Theorem 03Linear Functionals and the Riesz Representation Theorem Problem set0/10 Problem set 20/10

01Orthogonal Complements and the Direct Sum V = U ⊕ U⊥02Orthogonal Projection onto U and Why U⊥⊥ = U 03Riesz Representation Without a Basis: v ⊥ null φ Problem set0/10 Problem set 20/10

01The Closest Point Is the Projection: Fitting Sine on [−π, π]02The Pseudoinverse T† and Its Three Inverse-Like Properties 03T†w: The Best Approximate Solution of Smallest Norm Problem set0/10 Problem set 20/10

Self-Adjoint, Normal and Unitary Operators

01The Adjoint T*: Moving T into the Second Slot of ⟨Tv, w⟩02Null T* = (range T)⊥ and M(T*) as the Conjugate Transpose 03Self-Adjoint Operators: Real Eigenvalues and Real ⟨Tv, v⟩04Normal Operators: ‖Tv‖ = ‖T*v‖ and Orthogonal Eigenvectors Problem set0/10 Problem set 20/10

01Real Spectral Theorem: Self-Adjoint ⇔ Orthonormal Eigenbasis 02The Complex Spectral Theorem: Normal Is Enough 03Positive Operators: Six Equivalent Characterizations 04The Unique Positive Square Root √T Problem set0/10 Problem set 20/10

01Isometries: Five Equivalent Ways to Preserve Length 02Unitary Operators: S⁻¹ = S*, Eigenvalues on |λ| = 1 03QR Factorization: Gram–Schmidt Recorded as A = QR 04Cholesky Factorization: Positive Definite B = R*R Problem set0/10 Problem set 20/10

The Singular Value Decomposition

01Singular Values: Square Roots of the Eigenvalues of T*T 02Singular Value Decomposition: Rank, Isometries, Proof 03Diagonal Matrices, Adjoints and Pseudoinverses via SVD 04Matrix SVD: Computing It and Factoring A = BDC*Problem set0/10 Problem set 20/10

01The Norm of a Linear Map Is Its Largest Singular Value 02Computing the Norm of T: Three Formulas and Examples 03Best Low-Rank Approximation: Truncate the SVD at k 04Polar Decomposition: T = S√(T*T)Problem set0/10 Problem set 20/10

01Invertible Operators Turn Balls into Ellipsoids 02Boxes to Boxes: Parallelepipeds and the SVD 03Volume Scales by the Product of the Singular Values Problem set0/10 Problem set 20/10

Operators on Complex Vector Spaces

01Null Spaces of Powers Stabilize and Split the Space 02Generalized Eigenvectors Span Every Complex Space 03Linear Independence of Generalized Eigenvectors 04Nilpotent Operators: Zero Is the Only Eigenvalue Problem set0/10 Problem set 20/10

01Generalized Eigenspaces Split Every Complex Space 02Multiplicity Without Determinants: Algebraic vs Geometric 03Cayley–Hamilton Without Determinants 04Upper-Triangular Diagonals Count Multiplicities 05Block Diagonal Form: One Block per Eigenvalue Problem set0/10 Problem set 20/10

01Every Invertible Complex Operator Has a Square Root 02Jordan Bases: Strings and Ones Above the Diagonal 03Proving Jordan Form via an Invariant Complement 04The Trace Equals the Sum of the Eigenvalues 05Why ST − TS Can Never Be the Identity Problem set0/10 Problem set 20/10

Multilinear Algebra and Determinants

01Bilinear Forms: Linear in Each Slot, a Matrix per Basis 02Change of Basis for Bilinear Forms: A = CᵀBC 03Symmetric Bilinear Forms Are Diagonal in Some Basis 04Every Bilinear Form Splits: Symmetric + Alternating 05Quadratic Forms: Polarization and Sums of Squares Problem set0/10 Problem set 20/10

01Alternating Multilinear Forms: Dependent Lists Give Zero 02Swaps, Inversions and the Sign of a Permutation 03Top-Degree Alternating Forms: A One-Dimensional Space 04The Determinant of an Operator: αT = (det T)α 05The Determinant Formula: A Signed Sum over Permutations Problem set0/10 Problem set 20/10

01det(ST) = det S · det T: Invertibility and Eigenvalues 02Similarity Invariance: One Determinant in Every Basis 03det Aᵀ = det A: Row Operations and Gaussian Elimination 04|det T| Is Volume: Unitary, Positive and Singular Values 05The Characteristic Polynomial det(zI − T) over R and C 06Hadamard's Inequality: Column Lengths Cap |det A|07The Vandermonde Determinant via Polynomial Evaluation Problem set0/10 Problem set 20/10

Hadamard's Inequality: Column Lengths Cap |det A|

Why can no parallelepiped with given edge lengths beat the box, and how does QR factorization turn that into a bound on |det A|?

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Your summary note

1. Reducing to the QR factorization
  Dispose of the non-invertible case, then write $A = QR$ with $Q$ unitary and $R$ upper triangular with positive diagonal.
2. The chain of equalities and one inequality
  Write the full chain from $|\det A| = |\det R| = \prod R_{k,k}$ to $\prod \|v_k\|$ , marking the single inequality and the step where $Q$ preserves norms.
3. The geometric meaning
  Describe the image of the unit cube as the parallelepiped spanned by the columns and state which shape of parallelepiped has the largest volume for given edge lengths.
Attempt 1 of 2