 6.5.21E: Let A be an m × n matrix whose columns are linearly independent. [C...
 6.5.22E: Use Exercise 19 to show that rank ATA D rank A. [Hint: How many col...
 6.5.23E: Suppose A is m × n with linearly independent columns and b is in Rm...
 6.5.24E: Find a formula for the leastsquares solution of Ax = b when the co...
 6.5.25E: Describe all leastsquares solutions of the system
 6.5.1E: In Exercises 1–4, find a leastsquares solution of Ax = b by (a) co...
 6.5.2E: In Exercises 1–4, find a leastsquares solution of Ax = b by (a) co...
 6.5.3E: In Exercises 1–4, find a leastsquares solution of Ax = b by (a) co...
 6.5.4E: In Exercises 1–4, find a leastsquares solution of Ax = b by (a) co...
 6.5.5E: In Exercises 5 and 6, describe all leastsquares solutions of the e...
 6.5.6E: In Exercises 5 and 6, describe all leastsquares solutions of the e...
 6.5.7E: Compute the leastsquares error associated with the least squares s...
 6.5.8E: Compute the leastsquares error associated with the least squares s...
 6.5.9E: In Exercises 9–12, find (a) the orthogonal projection of b onto Col...
 6.5.10E: In Exercises 9–12, find (a) the orthogonal projection of b onto Col...
 6.5.11E: In Exercises 9–12, find (a) the orthogonal projection of b onto Col...
 6.5.12E: In Exercises 9–12, find (a) the orthogonal projection of b onto Col...
 6.5.13E: Compute Au and Av, and compare them with b. Could u possibly be a l...
 6.5.14E: Compute Au and Av, and compare them with b. Is it possible that at ...
 6.5.15E: In Exercises 15 and 16, use the factorization A = QR to find the le...
 6.5.16E: In Exercises 15 and 16, use the factorization A = QR to find the le...
 6.5.17E: In Exercises 17 and 18, A is an m × n matrix and b is in Rm. Mark e...
 6.5.18E: In Exercises 17 and 18, A is an m × n matrix and b is in Rm. Mark e...
 6.5.19E: Let A be an m × n matrix. Use the steps below to show that a vector...
 6.5.20E: Let A be an m × n matrix such that ATA is invertible. Show that the...
 6.5.26E: [M] Example 3 in Section 4.8 displayed a lowpass linear filter tha...
Solutions for Chapter 6.5: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 6.5
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Chapter 6.5 includes 26 full stepbystep solutions. Since 26 problems in chapter 6.5 have been answered, more than 47382 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

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.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

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