 5.1: Set x = [ 0 : 4, 4, 4, 1, 1] and y = ones(9, 1) (a) Use the MATLAB ...
 5.2: (Least Squares Fit to a Data Set by a Linear Function) The followin...
 5.3: (Least Squares Fit to a Data Set by a Linear Function) The followin...
 5.4: (Least Squares Circles) The parametric equations for a circle with ...
 5.5: (Fundamental Subspaces: Orthonormal Bases) The vector spaces N(A), ...
 5.6: True or FalseIf an m n matrix A has linearly dependent columnsand b...
 5.7: True or FalseIf N(A) = {0}, then the system Ax = b will have auniqu...
 5.8: True or FalseIf Q1 and Q2 are orthogonal matrices, then Q1Q2 alsois...
 5.9: True or FalseIf {u1, u2, ... , uk} is an orthonormal set of vectors...
 5.10: True or False. If {u1, u2, ... , uk} is an orthonormal set of vecto...
 5.11: The functions cos x and sin x are both unit vectors in C[, ] with i...
 5.12: The functions cos x and sin x are both unit vectors in C[, ] with i...
Solutions for Chapter 5: Orthogonality
Full solutions for Linear Algebra with Applications  9th Edition
ISBN: 9780321962218
Solutions for Chapter 5: Orthogonality
Get Full SolutionsChapter 5: Orthogonality includes 12 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Since 12 problems in chapter 5: Orthogonality have been answered, more than 10808 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Column space C (A) =
space of all combinations of the columns of A.

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.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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 •

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.

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.

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

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

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