 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 UpperTriangular 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: SelfAdjoint 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
ISBN: 9783319110790
Linear Algebra Done Right (Undergraduate Texts in Mathematics)  3rd Edition  Solutions by Chapter
Get Full SolutionsThis expansive textbook survival guide covers the following chapters: 31. Linear Algebra Done Right (Undergraduate Texts in Mathematics) was written by and is associated to the ISBN: 9783319110790. Since problems from 31 chapters in Linear Algebra Done Right (Undergraduate Texts in Mathematics) have been answered, more than 2819 students have viewed full stepbystep answer. The full stepbystep solution to problem in Linear Algebra Done Right (Undergraduate Texts in Mathematics) were answered by , our top Math solution expert on 03/15/18, 04:46PM. This textbook survival guide was created for the textbook: Linear Algebra Done Right (Undergraduate Texts in Mathematics), edition: 3.

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Norm
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 = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pivot.
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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , 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 = UI.
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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