 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 1847 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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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].

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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 matrix A.
A square matrix that has no inverse: det(A) = o.

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.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).
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