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

Operators on Inner Product Spaces

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

Norms, Orthogonality and the Pythagorean Theorem

How does an inner product define length and perpendicularity, and why does Pythagoras hold in every inner product space?

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Your summary note

1. The norm $\|v\| = \sqrt{\langle v, v\rangle}$ and $\|\lambda v\| = |\lambda|\,\|v\|$
  Define the norm, write it on $\mathbf{F}^n$ and for continuous functions on $[-1, 1]$ , and prove $\|\lambda v\| = |\lambda|\,\|v\|$ by expanding $\|\lambda v\|^2$ and taking square roots.
2. Orthogonal vectors and the vector $0$
  Define orthogonality by $\langle u, v\rangle = 0$ , record the $\mathbf{R}^2$ formula $\langle u, v\rangle = \|u\|\,\|v\|\cos\theta$ linking it to perpendicularity, and prove $0$ is the only self-orthogonal vector.
3. $\|u + v\|^2 = \|u\|^2 + \|v\|^2$ for orthogonal $u, v$
  Expand $\langle u + v, u + v\rangle$ into four terms and show both cross terms vanish when $\langle u, v\rangle = 0$ , including the conjugate-symmetry step for $\langle v, u\rangle$ .
Attempt 1 of 2