- 1.2.1: Which of the matrices that follow are in row echelon form? Which ar...
- 1.2.2: The augmented matrices that follow are in row echelon form. For eac...
- 1.2.3: The augmented matrices that follow are in reduced row echelon form....
- 1.2.4: For each of the systems in Exercise 3, make a list of the lead vari...
- 1.2.5: For each of the systems of equations that follow, use Gaussian elim...
- 1.2.6: Use GaussJordan reduction to solve each of the following systems. (...
- 1.2.7: Give a geometric explanation of why a homogeneous linear system con...
- 1.2.8: Consider a linear system whose augmented matrix is of the form 121 ...
- 1.2.9: Consider a linear system whose augmented matrix is of the form 12 1...
- 1.2.10: Consider a linear system whose augmented matrix is of the form 113 ...
- 1.2.11: Given the linear systems (i) x1 + 2x2 = 2 3x1 + 7x2 = 8 (ii) x1 + 2...
- 1.2.12: . Given the linear systems (i) x1 + 2x2 + x3 = 2 x1 x2 + 2x3 = 3 2x...
- 1.2.13: Given a homogeneous system of linear equations, if the system is ov...
- 1.2.14: Given a nonhomogeneous system of linear equations, if the system is...
- 1.2.15: Determine the values of x1, x2, x3, x4 for the following traffic fl...
- 1.2.16: Consider the traffic flow diagram that follows, where a1, a2, a3, a...
- 1.2.17: Let (c1, c2) be a solution of the 2 2 system a11x1 + a12x2 = 0 a21x...
- 1.2.18: In Application 3 the solution (6, 6, 6, 1) was obtained by setting ...
- 1.2.19: Liquid benzene burns in the atmosphere. If a cold object is placed ...
- 1.2.20: Nitric acid is prepared commercially by a series of three chemical ...
- 1.2.21: In Application 4, determine the relative values of x1, x2, and x3 i...
- 1.2.22: Determine the amount of each current for the following networks: (a...
Solutions for Chapter 1.2: Row Echelon Form
Full solutions for Linear Algebra with Applications | 9th Edition
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.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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].
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
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.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.