 4.6.1E: In Exercises 1–4, assume that the matrix A is row equivalent to B. ...
 4.6.2E: In Exercises 1–4, assume that the matrix A is row equivalent to B. ...
 4.6.3E: In Exercises 1–4, assume that the matrix A is row equivalent to B. ...
 4.6.4E: In Exercises 1–4, assume that the matrix A is row equivalent to B. ...
 4.6.5E: If a 4 × 7 matrix A has rank 3, find dim Nul A, dim Row A, and rank...
 4.6.6E: If a 7 × 5 matrix A has rank 2, find dim Nul A, dim Row A, and rank...
 4.6.7E: Suppose a 4 × 7 matrix A has four pivot columns. Is Col A = ? Is Nu...
 4.6.8E: Suppose a 6 × 8 matrix A has four pivot columns. What is dim Nul A?...
 4.6.9E: If the null space of a 4 × 6 matrix A is 3dimensional, what is the...
 4.6.10E: If the null space of an 8 × 7 matrix A is 5dimensional, what is th...
 4.6.11E: If the null space of an 8 × 5 matrix A is 3dimensional, what is th...
 4.6.12E: If the null space of a 5 × 4 matrix A is 2dimensional, what is the...
 4.6.13E: If A is a 7 × 5 matrix, what is the largest possible rank of A?If A...
 4.6.14E: If A is a 5 × 4 matrix, what is the largest possible dimension of t...
 4.6.15E: If A is a 3 × 7 matrix, what is the smallest possible dimension of ...
 4.6.16E: If A is a 7 × 5 matrix, what is the smallest possible dimension of ...
 4.6.17E: In Exercises 17 and 18, A is an m × n matrix. Mark each statement T...
 4.6.18E: In Exercises 17 and 18, A is an m × n matrix. Mark each statement T...
 4.6.19E: Suppose the solutions of a homogeneous system of five linear equati...
 4.6.20E: Suppose a nonhomogeneous system of six linear equations in eight un...
 4.6.21E: Suppose a nonhomogeneous system of nine linear equations in ten unk...
 4.6.22E: Is is possible that all solutions of a homogeneous system of ten li...
 4.6.23E: A homogeneous system of twelve linear equations in eight unknowns h...
 4.6.24E: Is it possible for a nonhomogeneous system of seven equations in si...
 4.6.25E: A scientist solves a nonhomogeneous system of ten linear equations ...
 4.6.26E: In statistical theory, a common requirement is that a matrix be of ...
 4.6.27E: Exercises 27–29 concern an m × n matrix A and what are often called...
 4.6.28E: Exercises 27–29 concern an m × n matrix A and what are often called...
 4.6.29E: Exercises 27–29 concern an m × n matrix A and what are often called...
 4.6.30E: Suppose A is m × n and b is in . What has to be true about the two ...
 4.6.31E: Rank 1 matrices are important in some computer algorithms and sever...
 4.6.32E: Rank 1 matrices are important in some computer algorithms and sever...
 4.6.33E: Rank 1 matrices are important in some computer algorithms and sever...
 4.6.34E: Let A be an m ? n matrix of rank r > 0 and let U be an echelon form...
 4.6.35E: a. Construct matrices C and N whose columns are bases for Col A and...
 4.6.36E: [M] Repeat Exercise 35 for a random integervalued 6 × 7 matrix A w...
 4.6.37E: [M] Let A be the matrix in Exercise 35. Construct a matrix C whose ...
 4.6.38E: [M] Repeat Exercise 37 for three random integervalued 5 × 7 matric...
Solutions for Chapter 4.6: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 4.6
Get Full SolutionsSince 38 problems in chapter 4.6 have been answered, more than 35352 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Chapter 4.6 includes 38 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Solvable system Ax = b.
The right side b is in the column space of A.

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.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.