- Chapter 1.A: Rn and Cn
- Chapter 1.B: Definition of Vector Space
- Chapter 1.C: Subspaces
- Chapter 10.A: Trace
- Chapter 10.B: Determinant
- Chapter 2.A: Span and Linear Independence
- Chapter 2.B: Bases
- Chapter 2.C: Dimension
- Chapter 3.A: The Vector Space of Linear Maps
- Chapter 3.B: Null Spaces and Ranges
- Chapter 3.C: Matrices
- Chapter 3.D: Invertibility and Isomorphic Vector Spaces
- Chapter 3.E: Products and Quotients of Vector Spaces
- Chapter 3.F: Duality
- Chapter 4: Polynomials
- Chapter 5.A: Invariant Subspaces
- Chapter 5.B: Eigenvectors and Upper-Triangular Matrices
- Chapter 5.C: Eigenspaces and Diagonal Matrices
- Chapter 6.A: Inner Products and Norms
- Chapter 6.B: Orthonormal Bases
- Chapter 6.C: Orthogonal Complements and Minimization Problems
- Chapter 7.A: Self-Adjoint and Normal Operators
- Chapter 7.B: The Spectral Theorem
- Chapter 7.C: Positive Operators and Isometries
- Chapter 7.D: Polar Decomposition and Singular Value Decomposition
- Chapter 8.A: Generalized Eigenvectors and Nilpotent Operators
- Chapter 8.B: Decomposition of an Operator
- Chapter 8.C: Characteristic and Minimal Polynomials
- Chapter 8.D: Jordan Form
- Chapter 9.A: Complexification
- Chapter 9.B: Operators on Real Inner Product Spaces
Linear Algebra Done Right (Undergraduate Texts in Mathematics) 3rd Edition - Solutions by Chapter
Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics) | 3rd Edition
Linear Algebra Done Right (Undergraduate Texts in Mathematics) | 3rd Edition - Solutions by ChapterGet Full Solutions
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Invert A by row operations on [A I] to reach [I A-I].
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
A directed graph that has constants Cl, ... , Cm associated with the edges.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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