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

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

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

Singular Values: Square Roots of the Eigenvalues of T*T

Why are the eigenvalues of T*T never negative, and what do their square roots reveal that T's own eigenvalues miss?

Loading…

←Previous Cholesky Factorization: Positive Definite B = R*R Next Singular Value Decomposition: Rank, Isometries, Proof →

Your summary note

1. Four properties of $T^*T$
  State the four properties for $T \in \mathcal{L}(V,W)$ and prove each: self-adjointness with $\langle T^*Tv, v\rangle = \|Tv\|^2$ , both null-space inclusions, the range equality, and the equal range dimensions.
2. Definition of the singular values of $T$
  Write the definition: nonnegative square roots of the eigenvalues of $T^*T$ , each repeated by the dimension of its eigenspace, listed in decreasing order, and note that positivity of $T^*T$ makes them defined.
3. Worked example on $\mathbf{F}^4$ where eigenvalues miss $2$
  For $T(z_1,z_2,z_3,z_4) = (0, 3z_1, 2z_2, -3z_4)$ compute $T^*T$ and its eigenspace dimensions, obtain singular values $3, 3, 2, 0$ , and record that the eigenvalues of $T$ are only $-3$ and $0$ .
4. Example from $\mathbf{F}^4$ to $\mathbf{F}^3$ and the list length
  From the given $3$ -by- $4$ matrix compute the matrix of $T^*T$ and its eigenspace dimensions, obtain singular values $5, \sqrt{2}, 0, 0$ , and record that the list length always equals $\dim V$ .
Attempt 1 of 2