 2.1.1: Find a row echelon fonn of each of the given matrices. Record the r...
 2.1.2: Find a row echelon fonn of each of the given matrices. Record the r...
 2.1.3: Each of the given matrices is in row echelon fonn. Delermine its re...
 2.1.4: Each of the given matrices is in row echelon fonn. De5. tennine its...
 2.1.5: Find the reduced row echelon fonn of each of the given matrices. Re...
 2.1.6: Find the reduced row echelon fonn of each of the given matrices. Re...
 2.1.7: Let x. y . z. and w be nonzero real numbers. Label each of Ihe foll...
 2.1.8: Let x . y . ; . and w be nonzero rt!al numbers. Label each of the f...
 2.1.9: Let A be an /I x II matrix in reduced row echelon form. [ cosO A~ S...
 2.1.10: Prove: (a) Every matrix is row equivalent to itself. (b) If B is ro...
 2.1.11: (a) Find a matrix in column echelon form that is column equivalent ...
 2.1.12: Repeat Exercise II for the matrix 2 3 4 3  I 2 4 lJ
 2.1.13: Determine the reduced row echelon form of
Solutions for Chapter 2.1: Echelon Form of a Matrix
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 2.1: Echelon Form of a Matrix
Get Full SolutionsChapter 2.1: Echelon Form of a Matrix includes 13 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 13 problems in chapter 2.1: Echelon Form of a Matrix have been answered, more than 9193 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

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

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.

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

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.