 5.6.1: For each of the following, use the GramSchmidt process to find an o...
 5.6.2: Factor each of the matrices in Exercise 1 into a product QR, where ...
 5.6.3: Given the basis {(1, 2,2)T , (4, 3, 2)T , (1, 2, 1)T } for R3, use ...
 5.6.4: Consider the vector space C[1, 1] with inner product defined by _ f...
 5.6.5: Let A = 2 1 1 1 2 1 and b = 12 6 18 (a) Use the GramSchmidt process...
 5.6.6: Repeat Exercise 5, using A = 3 1 4 2 0 2 and b = 0 20 10
 5.6.7: The vectors x1 = 1 2 (1, 1, 1,1)T and x2 = 1 6 (1, 1, 3, 5)T form a...
 5.6.8: Use the GramSchmidt process to find an orthonormal basis for the su...
 5.6.9: Repeat Exercise 8, using the modified Gram Schmidt process. Compare...
 5.6.10: Let A be an m 2 matrix. Show that if both the classical GramSchmidt...
 5.6.11: Let A be an m 3 matrix. Let QR be the QR factorization obtained whe...
 5.6.12: What will happen if the GramSchmidt process is applied to a set of ...
 5.6.13: Let A be an m n matrix of rank n and let b Rm. Show that if Q and R...
 5.6.14: Let U be an mdimensional subspace of Rn and let V be a kdimension...
 5.6.15: Dimension Theorem Let U and V be subspaces of Rn. In the case that ...
Solutions for Chapter 5.6: The GramSchmidt Orthogonalization Process
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 5.6: The GramSchmidt Orthogonalization Process
Get Full SolutionsSince 15 problems in chapter 5.6: The GramSchmidt Orthogonalization Process have been answered, more than 5041 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.6: The GramSchmidt Orthogonalization Process includes 15 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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 .

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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