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

Quadratic Forms: Polarization and Sums of Squares

How does polarization recover a symmetric form from its quadratic form, and why does some basis turn every quadratic form into a weighted sum of squares?

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Your summary note

1. Quadratic forms $q_\beta(v) = \beta(v,v)$ on $\mathbf{F}^n$
  Define $q_\beta$ , prove that $q$ is quadratic on $\mathbf{F}^n$ exactly when $q$ , and record that $q_\beta = 0$ exactly when $\beta$ is alternating.
2. Unique symmetric form and the polarization test
  Prove each quadratic form is $q_\rho$ for a unique symmetric $\rho$ , and prove the test: $q(2v) = 4q(v)$ with $q(u+w) - q(u) - q(w)$ symmetric bilinear, recovering $\rho$ by halving it.
3. Symmetrizing $x_1^2 - 4x_1x_2 + 8x_1x_3 - 3x_3^2$
  Split each cross-term coefficient evenly between $x_jy_k$ and $x_ky_j$ to write the symmetric $\rho$ for this $q$ , and check that $\rho(v,v) = q(v)$ .
4. Diagonalizing a quadratic form into weighted squares
  Take a basis diagonalizing $\rho$ , expand $q(x_1e_1 + \cdots + x_ne_n)$ to $\lambda_1 x_1^2 + \cdots + \lambda_n x_n^2$ , and note the basis can be orthonormal on a real inner product space.
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