 6.2.1: babel the following statements as true or false. (a) The Gram Schmi...
 6.2.2: In each pari, apply the Gram Schmidt process to the given subset S ...
 6.2.3: In R2 , lei "((4= . 4= Find (he Fourier coefficients of (3,4) rela...
 6.2.4: Let S = {(1,0,i), (1,2,1)} in C3 . Compute S1
 6.2.5: Let S ./u} where XQ is a nonzero vector in R,{ . Describe Sn geome...
 6.2.6: Let V be an inner product space, and let W be a finitedimensional ...
 6.2.7: Let 0 be a basis lor a subspace W of an inner product space V. and ...
 6.2.8: Prove that if { a\. w2 ir } is an orthogonal set of nonzero vector...
 6.2.9: Let W  span({(/'.(). 1)}) in C:{ . Find orthonormal bases for W an...
 6.2.10: Let W be a. finitedimensional subspace of an inner product space V...
 6.2.11: Let A be an // x /; matrix with complex entries. Prove thai .LP = /...
 6.2.12: Prove that for any matrix A ; n (R(U.)r =N(L,
 6.2.13: Let V be an inner product space, S and SQ be subsets of V. and W be...
 6.2.14: Let Wi and W2 be subspaces of a. finitedimensional inner product s...
 6.2.15: Let V be a finitedimensional inner product space over F. (a) Parse...
 6.2.16: a) Bessel's Inequality. Let V be an inner product space, and let S ...
 6.2.17: Let T be a linear operator on an inner product space V. If (T(x),y)...
 6.2.18: Let V = C([l, 1]). Suppose that W,. and W denote the subspaces of ...
 6.2.19: In each of the following parts, find the orthogonal projection of t...
 6.2.20: In each part of Exercise 19, find the distance from the given vecto...
 6.2.21: Let V = C([l, 1]) with the inner product (f,g) = f \ f(t)g(t)dt, a...
 6.2.22: Let V = C([0,1]) with the inner product. (f,g) = /J f(t)g(t) dt. Le...
 6.2.23: Let V be the vector space defined in Example 5 of Section 1.2, the ...
Solutions for Chapter 6.2: GramSchmidt Orthogonalization Process
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 6.2: GramSchmidt Orthogonalization Process
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra was written by and is associated to the ISBN: 9780130084514. Chapter 6.2: GramSchmidt Orthogonalization Process includes 23 full stepbystep solutions. Since 23 problems in chapter 6.2: GramSchmidt Orthogonalization Process have been answered, more than 10870 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra , edition: 4.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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 .

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.