 5.5.1: Let W be the subspace of R.1 spanned by the vector (3) Find a basis...
 5.5.2: Let IV ~ 'P'" I [J . [n I (3) Find a basis for W.i. (b ) Describe ...
 5.5.3: Let \V be the subspace of Rs spanned by the vectors 11',. W 1. 11...
 5.5.4: Let \V be the subspace of R4 spanned by the vectors 11'1. W 1. 11...
 5.5.5: (Ca lclllus Required). Let V be the Euclidean space 1'3 with the in...
 5.5.6: Let V be the Euclidean space 1'4 with the inner proouct defined in ...
 5.5.7: Let W be the plane 3.r +2)'  ~ =0 in R}. Find a basis for W.i.
 5.5.8: Let V be the Euclidean space of all 2 x 2 matrices with the inner p...
 5.5.9: III Exaci!le!l 9 alld 10, cOlllpllfe Ihe jOllr fillldwlIl'lila/ll'c...
 5.5.10: III Exaci!le!l 9 alld 10, cOlllpllfe Ihe jOllr fillldwlIl'lila/ll'c...
 5.5.11: III ExerciJe.l II throllgh /4,jilld projwv jor Ihe gil'ell I'ector ...
 5.5.12: III ExerciJe.l II throllgh /4,jilld projwv jor Ihe gil'ell I'ector ...
 5.5.13: Let V be the vector space of real valued continuous func tions on [...
 5.5.14: Let IV be the plane in R3 given by the equation x+ )" 2:;:=0.
 5.5.15: Let IV be the subspncc of Rl wilh ol1hononnnl basis [WI. W2\, where...
 5.5.16: Let \V be the subspace of R4 with otthonormal basis [WI. ',1,'2. W3...
 5.5.17: Let IV be the subspace of continuous functions on [ Jr. It] define...
 5.5.18: Let IV be the plane in R3 given by the equation x  )"  , ~ o. W,;...
 5.5.19: Let IV be the subspace of Rl defined in Exercise 15. and
 5.5.20: Let IV be the subspace of R4 defined in Exercise 16. and ,,, ' ~ [ ...
 5.5.21: Let IV be the subspace of continuous functions on [ J"[. It[ defin...
 5.5.22: III Exercise.l 22 {llId 23, find the f"tJllrier polynomial of degre...
 5.5.23: III Exercise.l 22 {llId 23, find the f"tJllrier polynomial of degre...
 5.5.24: Show that if V is an itmer product space and IV is II sub space of ...
 5.5.25: Let V be an inner product space. Show that the otthog. onal complem...
 5.5.26: Show Ihal if IV is a Hlbspace of an inner product . pace V that is ...
 5.5.27: Let A be an //I x II matrix. Show that every vector \' in R" can be...
 5.5.28: Let V be a Euclidean space, and IV a sub.>pace of V. Show that if W...
 5.5.29: Let IV be a subspace of an inner product space V and let {WI. w, .....
Solutions for Chapter 5.5: Orthogonal Complements
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 5.5: Orthogonal Complements
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.5: Orthogonal Complements includes 29 full stepbystep solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. Since 29 problems in chapter 5.5: Orthogonal Complements have been answered, more than 13761 students have viewed full stepbystep solutions from this chapter.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.