 4.7.1: LetA~ ~  ; ~]  8 4 8 (a) Find the set of all solutiorls to Ax = O...
 4.7.2: Let (a) Find the set of all solutions to Ax = O.(b) Express each so...
 4.7.3: III t.xerdl'es J thmugh 10. filld a Ixt.I'isj(Jr and Ihe dimellsion...
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 4.7.7: III t.xerdl'es J thmugh 10. filld a Ixt.I'isj(Jr and Ihe dimellsion...
 4.7.8: III t.xerdl'es J thmugh 10. filld a Ixt.I'isj(Jr and Ihe dimellsion...
 4.7.9: III t.xerdl'es J thmugh 10. filld a Ixt.I'isj(Jr and Ihe dimellsion...
 4.7.10: III t.xerdl'es J thmugh 10. filld a Ixt.I'isj(Jr and Ihe dimellsion...
 4.7.11: III IelL'i.1'e.f II alld 12. find a basilfor Ihe null .1'pace of e...
 4.7.12: III IelL'i.1'e.f II alld 12. find a basilfor Ihe null .1'pace of e...
 4.7.13: III tercise.\' I J Ihmllgh 16. find a basi.I' for Ihe SOllilioll jp...
 4.7.14: III tercise.\' I J Ihmllgh 16. find a basi.I' for Ihe SOllilioll jp...
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 4.7.16: III tercise.\' I J Ihmllgh 16. find a basi.I' for Ihe SOllilioll jp...
 4.7.17: III Exerci.l'es 17 Ihmugh 20. find all real number.I' A .l'IIch tha...
 4.7.18: III Exerci.l'es 17 Ihmugh 20. find all real number.I' A .l'IIch tha...
 4.7.19: III Exerci.l'es 17 Ihmugh 20. find all real number.I' A .l'IIch tha...
 4.7.20: III Exerci.l'es 17 Ihmugh 20. find all real number.I' A .l'IIch tha...
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 4.7.22: III Exerci.l'e.l" 21 alld 22, delennine Ihe .I'Ollilioll 10 Ihe IiI...
 4.7.23: LeI S = {XI . X! ..... "d be a set of solutions 10 a homogeneous s...
 4.7.24: Show that if the 1/ x 11 coefficient matrix A of the homogeneous sy...
 4.7.25: (a ) Show that the zero matrix is the only 3 x 3 matrix whose null ...
 4.7.26: Matrices A and B are III x I!. and their reduced row echelon fomls ...
Solutions for Chapter 4.7: Homogeneous Systems
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 4.7: Homogeneous Systems
Get Full SolutionsElementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. Since 26 problems in chapter 4.7: Homogeneous Systems have been answered, more than 12033 students have viewed full stepbystep solutions from this chapter. Chapter 4.7: Homogeneous Systems includes 26 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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].

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

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.

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

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

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.

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 •

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

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.

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