 5.5.5.1: G is a generator matrix for a code C. Bring G into standard form an...
 5.5.1.1: determine which sets of vectors are orthogonal.
 5.5.2.1: find the orthogonal complement W of W and give a basis for W. W e c...
 5.5.3.1: the given vectors form a basis for 2 or 3 . Apply the GramSchmidt ...
 5.5.4.1: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.2: G is a generator matrix for a code C. Bring G into standard form an...
 5.5.1.2: determine which sets of vectors are orthogonal.
 5.5.2.2: find the orthogonal complement W of W and give a basis for W. W e c...
 5.5.3.2: the given vectors form a basis for 2 or 3 . Apply the GramSchmidt ...
 5.5.4.2: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.3: G is a generator matrix for a code C. Bring G into standard form an...
 5.5.1.3: determine which sets of vectors are orthogonal.
 5.5.2.3: find the orthogonal complement W of W and give a basis for W. W xyz...
 5.5.3.3: the given vectors form a basis for 2 or 3 . Apply the GramSchmidt ...
 5.5.4.3: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.4: G is a generator matrix for a code C. Bring G into standard form an...
 5.5.1.4: determine which sets of vectors are orthogonal.
 5.5.2.4: find the orthogonal complement W of W and give a basis for W. W xyz...
 5.5.3.4: the given vectors form a basis for 2 or 3 . Apply the GramSchmidt ...
 5.5.4.4: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.5: P is a parity check matrix for a code C. Bring P into standard form...
 5.5.1.5: determine which sets of vectors are orthogonal.
 5.5.2.5: find the orthogonal complement W of W and give a basis for W.W xyz ...
 5.5.3.5: the given vectors form a basis for a subspace W of 3 or 4 . Apply t...
 5.5.4.5: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.6: P is a parity check matrix for a code C. Bring P into standard form...
 5.5.1.6: determine which sets of vectors are orthogonal.
 5.5.2.6: find the orthogonal complement W of W and give a basis for W. W xyz...
 5.5.3.6: the given vectors form a basis for a subspace W of 3 or 4 . Apply t...
 5.5.4.6: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.7: P is a parity check matrix for a code C. Bring P into standard form...
 5.5.1.7: show that the given vectors form an orthogonal basis for 2 or 3 . T...
 5.5.2.7: find bases for the row space and null space of A. Verify that every...
 5.5.3.7: find the orthogonal decomposition of v with respect to the subspace W.
 5.5.4.7: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.8: P is a parity check matrix for a code C. Bring P into standard form...
 5.5.1.8: show that the given vectors form an orthogonal basis for 2 or 3 . T...
 5.5.2.8: find bases for the row space and null space of A. Verify that every...
 5.5.3.8: find the orthogonal decomposition of v with respect to the subspace W.
 5.5.4.8: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.9: find the dual code C of the code C.
 5.5.1.9: show that the given vectors form an orthogonal basis for 2 or 3 . T...
 5.5.2.9: find bases for the column space of A and the null space of AT for t...
 5.5.3.9: Use the GramSchmidt Process to find an orthogonal basis for the co...
 5.5.4.9: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.10: find the dual code C of the code C.
 5.5.1.10: show that the given vectors form an orthogonal basis for 2 or 3 . T...
 5.5.2.10: find bases for the column space of A and the null space of AT for t...
 5.5.3.10: Use the GramSchmidt Process to find an orthogonal basis for the co...
 5.5.4.10: Orthogonally diagonalize the matrices in Exercises 110 by finding a...
 5.5.5.11: find the dual code C of the code C.
 5.5.1.11: determine whether the given orthogonal set of vectors is orthonorma...
 5.5.2.11: let W be the subspace spanned by the given vectors. Find a basis fo...
 5.5.3.11: Find an orthogonal basis for 3 that contains the vector
 5.5.4.11: If b 0, orthogonally diagonalize A ca bb a d
 5.5.5.12: find the dual code C of the code C.
 5.5.1.12: determine whether the given orthogonal set of vectors is orthonorma...
 5.5.2.12: let W be the subspace spanned by the given vectors. Find a basis fo...
 5.5.3.12: Find an orthogonal basis for 4 that contains the vectors 2 1 0 1 an...
 5.5.4.12: If b 0, orthogonally diagonalize A a 0 b0 a 0b 0 a
 5.5.5.13: either a generator matrix G or a parity check matrix P is given for...
 5.5.1.13: determine whether the given orthogonal set of vectors is orthonorma...
 5.5.2.13: let W be the subspace spanned by the given vectors. Find a basis fo...
 5.5.3.13: fill in the missing entries of Q to make Q an orthogonal matrix.
 5.5.4.13: Let A and B be orthogonally diagonalizable n n matrices and let c b...
 5.5.5.14: either a generator matrix G or a parity check matrix P is given for...
 5.5.1.14: determine whether the given orthogonal set of vectors is orthonorma...
 5.5.2.14: let W be the subspace spanned by the given vectors. Find a basis fo...
 5.5.3.14: fill in the missing entries of Q to make Q an orthogonal matrix.
 5.5.4.14: If A is an invertible matrix that is orthogonally diagonalizable, s...
 5.5.5.15: either a generator matrix G or a parity check matrix P is given for...
 5.5.1.15: determine whether the given orthogonal set of vectors is orthonorma...
 5.5.2.15: find the orthogonal projection of v onto the subspace W spanned by ...
 5.5.3.15: find a QR factorization of the matrixin the given exercise.Exercise 9
 5.5.4.15: If A and B are orthogonally diagonalizable and AB BA, show that AB ...
 5.5.5.16: either a generator matrix G or a parity check matrix P is given for...
 5.5.1.16: determine whether the given matrix is orthogonal. If it is, find it...
 5.5.2.16: find the orthogonal projection of v onto the subspace W spanned by ...
 5.5.3.16: find a QR factorization of the matrixin the given exercise.Exercise 10
 5.5.4.16: If A is a symmetric matrix, show that every eigenvalue of A is nonn...
 5.5.5.17: Find generator and parity check matrices for the dual of the (7, 4)...
 5.5.1.17: determine whether the given matrix is orthogonal. If it is, find it...
 5.5.2.17: find the orthogonal projection of v onto the subspace W spanned by ...
 5.5.3.17: he columns of Q were obtained byapplying the GramSchmidt Process t...
 5.5.4.17: find a spectral decomposition of the matrix in the given exercise. ...
 5.5.5.18: (a) Find generator and parity check matrices for E3 and Rep3. (b) S...
 5.5.1.18: determine whether the given matrix is orthogonal. If it is, find it...
 5.5.2.18: find the orthogonal projection of v onto the subspace W spanned by ...
 5.5.3.18: he columns of Q were obtained byapplying the GramSchmidt Process t...
 5.5.4.18: find a spectral decomposition of the matrix in the given exercise. ...
 5.5.5.19: Show that En and Repn are dual to each other
 5.5.1.19: determine whether the given matrix is orthogonal. If it is, find it...
 5.5.2.19: find the orthogonal decomposition of v with respect to W
 5.5.3.19: If A is an orthogonal matrix, find a QR factorization of A.
 5.5.4.19: find a spectral decomposition of the matrix in the given exercise.E...
 5.5.5.20: If C and D are codes and C D, show that D C.
 5.5.1.20: determine whether the given matrix is orthogonal. If it is, find it...
 5.5.2.20: find the orthogonal decomposition of v with respect to W
 5.5.3.20: Prove that A is invertible if and only if A QR, where Q is orthogon...
 5.5.4.20: find a spectral decomposition of the matrix in the given exercise.E...
 5.5.5.21: Show that if C is a code with a generator matrix, then (C) C.
 5.5.1.21: determine whether the given matrix is orthogonal. If it is, find it...
 5.5.2.21: find the orthogonal decomposition of v with respect to W
 5.5.3.21: use the method suggested by Exercise 20 to compute A1 for the matri...
 5.5.4.21: find a symmetric 2 2 matrix with eigenvalues l1 and l2 and correspo...
 5.5.5.22: Find a self dual code of length 6.
 5.5.1.22: Prove Theorem 5.8(a).
 5.5.2.22: find the orthogonal decomposition of v with respect to W
 5.5.3.22: use the method suggested by Exercise 20 to compute A1 for the matri...
 5.5.4.22: find a symmetric 2 2 matrix with eigenvalues l1 and l2 and correspo...
 5.5.5.23: evaluate the quadratic form f(x) xT Ax for the given A and x.
 5.5.1.23: Prove Theorem 5.8(b).
 5.5.2.23: Prove Theorem 5.9(c).
 5.5.3.23: Let A be an m n matrix with linearly independent columns. Give an a...
 5.5.4.23: find a symmetric 3 3 matrix witheigenvalues l1, l2, and l3 and corr...
 5.5.5.24: evaluate the quadratic form f(x) xT Ax for the given A and x.
 5.5.1.24: Prove Theorem 5.8(d).
 5.5.2.24: Prove Theorem 5.9(d).
 5.5.3.24: Let A be an m n matrix with linearly independent columns and let A ...
 5.5.4.24: find a symmetric 3 3 matrix witheigenvalues l1, l2, and l3 and corr...
 5.5.4.25: Let q be a unit vector in n and let W be the subspace spanned by q....
 5.5.5.25: evaluate the quadratic form f(x) xT Ax for the given A and x.
 5.5.1.25: Prove that every permutation matrix is orthogonal.
 5.5.2.25: Let W be a subspace of n and v a vector in n . Suppose that w and w...
 5.5.4.26: Let {q1,..., qk} be an orthonormal set of vectors in n and let W be...
 5.5.5.26: evaluate the quadratic form f(x) xT Ax for the given A and x.
 5.5.1.26: If Q is an orthogonal matrix, prove that any matrix obtained by rea...
 5.5.2.26: Let {v1,..., vn} be an orthogonal basis for n and let W span(v1,......
 5.5.4.27: Let A be an n n real matrix, all of whose eigenvalues are real. Pro...
 5.5.5.27: evaluate the quadratic form f(x) xT Ax for the given A and x.
 5.5.1.27: Let Q be an orthogonal 2 2 matrix and let x and y be vectors in 2 ....
 5.5.2.27: let W be a subspace of n , and let x be a vector in n . 2Prove that...
 5.5.4.28: Let A be a nilpotent matrix (see Exercise 56 in Section 4.2). Prove...
 5.5.5.28: evaluate the quadratic form f(x) xT Ax for the given A and x.
 5.5.1.28: (a) Prove that an orthogonal 2 2 matrix must have the form where is...
 5.5.2.28: let W be a subspace of n , and let x be a vector in n . 3. Prove th...
 5.5.5.29: find the symmetric matrix A associated with the given quadratic form.
 5.5.1.29: use Exercise 28 to determine whether the given orthogonal matrix re...
 5.5.2.29: let W be a subspace of n , and let x be a vector in n . 4 Prove tha...
 5.5.5.30: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.1.30: use Exercise 28 to determine whether the given orthogonal matrix re...
 5.5.2.30: Let be an orthonormal set in n , and let x be a vector in n . (a) P...
 5.5.5.31: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.1.31: use Exercise 28 to determine whether the given orthogonal matrix re...
 5.5.5.32: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.1.32: use Exercise 28 to determine whether the given orthogonal matrix re...
 5.5.5.33: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.1.33: Let A and B be orthogonal matrices. (a) Prove that . (b) Use part (...
 5.5.5.34: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.1.34: Let x be a unit vector in n . Partition x as Let Prove that Q is or...
 5.5.5.35: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.1.35: Prove that if an upper triangular matrix is orthogonal, then it mus...
 5.5.5.36: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.1.36: Prove that if , then there is no matrix A such that for all x in n .
 5.5.5.37: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.1.37: Let B be an orthonormal basis for n . (a) Prove that, for any x and...
 5.5.5.38: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.5.39: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.5.40: Diagonalize the quadratic forms in Exercises 3540 by finding an ort...
 5.5.5.41: Classify each of the quadratic forms in Exercises 4148 as positive ...
 5.5.5.42: Classify each of the quadratic forms in Exercises 4148 as positive ...
 5.5.5.43: Classify each of the quadratic forms in Exercises 4148 as positive ...
 5.5.5.44: Classify each of the quadratic forms in Exercises 4148 as positive ...
 5.5.5.45: Classify each of the quadratic forms in Exercises 4148 as positive ...
 5.5.5.46: Classify each of the quadratic forms in Exercises 4148 as positive ...
 5.5.5.47: Classify each of the quadratic forms in Exercises 4148 as positive ...
 5.5.5.48: Classify each of the quadratic forms in Exercises 4148 as positive ...
 5.5.5.49: . Prove Theorem 5.24.
 5.5.5.50: Let be a symmetric 2 2 matrix. Prove that A is positive definite if...
 5.5.5.51: Let B be an invertible matrix. Show that A BT B is
 5.5.5.52: Let A be a positive definite symmetric matrix. Show that there exis...
 5.5.5.53: Let A and B be positive definite symmetric n n matrices and let c b...
 5.5.5.54: Let A be a positive definite symmetric matrix. Show that there is a...
 5.5.5.55: find the maximum and minimum values of the quadratic form f(x) in t...
 5.5.5.56: find the maximum and minimum values of the quadratic form f(x) in t...
 5.5.5.57: find the maximum and minimum values of the quadratic form f(x) in t...
 5.5.5.58: find the maximum and minimum values of the quadratic form f(x) in t...
 5.5.5.59: Finish the proof of Theorem 5.25(a).
 5.5.5.60: Prove Theorem 5.25(c).
 5.5.5.61: identify the graph of the given equation.
 5.5.5.62: identify the graph of the given equation.
 5.5.5.63: identify the graph of the given equation.
 5.5.5.64: identify the graph of the given equation.
 5.5.5.65: identify the graph of the given equation.
 5.5.5.66: identify the graph of the given equation.
 5.5.5.67: use a translation of axes to put the conic in standard position. Id...
 5.5.5.68: use a translation of axes to put the conic in standard position. Id...
 5.5.5.69: use a translation of axes to put the conic in standard position. Id...
 5.5.5.70: use a translation of axes to put the conic in standard position. Id...
 5.5.5.71: use a translation of axes to put the conic in standard position. Id...
 5.5.5.72: use a translation of axes to put the conic in standard position. Id...
 5.5.5.73: use a rotation of axes to put the conic in standard position. Ident...
 5.5.5.74: use a rotation of axes to put the conic in standard position. Ident...
 5.5.5.75: use a rotation of axes to put the conic in standard position. Ident...
 5.5.5.76: use a rotation of axes to put the conic in standard position. Ident...
 5.5.5.77: identify the conic with the given equation and give its equation in...
 5.5.5.78: identify the conic with the given equation and give its equation in...
 5.5.5.79: identify the conic with the given equation and give its equation in...
 5.5.5.80: identify the conic with the given equation and give its equation in...
 5.5.5.81: Sometimes the graph of a quadratic equation is a straight line, a p...
 5.5.5.82: Sometimes the graph of a quadratic equation is a straight line, a p...
 5.5.5.83: Sometimes the graph of a quadratic equation is a straight line, a p...
 5.5.5.84: Sometimes the graph of a quadratic equation is a straight line, a p...
 5.5.5.85: Sometimes the graph of a quadratic equation is a straight line, a p...
 5.5.5.86: Sometimes the graph of a quadratic equation is a straight line, a p...
 5.5.5.87: Let A be a symmetric 2 2 matrix and let k be a scalar. Prove that t...
 5.5.5.88: identify the quadric with the given equation and give its equation ...
 5.5.5.89: identify the quadric with the given equation and give its equation ...
 5.5.5.90: identify the quadric with the given equation and give its equation ...
 5.5.5.91: identify the quadric with the given equation and give its equation ...
 5.5.5.92: identify the quadric with the given equation and give its equation ...
 5.5.5.93: identify the quadric with the given equation and give its equation ...
 5.5.5.94: identify the quadric with the given equation and give its equation ...
 5.5.5.95: identify the quadric with the given equation and give its equation ...
 5.5.5.96: Let A be a real matrix with complex eigenvalues such that and . Pro...
Solutions for Chapter 5: Orhthogonality
Full solutions for Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign)  3rd Edition
ISBN: 9780538735452
Solutions for Chapter 5: Orhthogonality
Get Full SolutionsChapter 5: Orhthogonality includes 215 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) was written by and is associated to the ISBN: 9780538735452. Since 215 problems in chapter 5: Orhthogonality have been answered, more than 13527 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign), edition: 3.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of 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.

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

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

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