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# 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

Solutions for Chapter 5.6
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##### ISBN: 9780136009290

Since 15 problems in chapter 5.6: The GramSchmidt Orthogonalization Process have been answered, more than 5041 students have viewed full step-by-step 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 step-by-step 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.

Key Math Terms and definitions covered in this textbook
• 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 n-l - 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 edge-node 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.