 1.5.1: Decide whether each matrix below is an elementary matrix. (a) (b) (...
 1.5.2: Decide whether each matrix below is an elementary matrix. (a) (b) (...
 1.5.3: Find a row operation and the corresponding elementry matrix that wi...
 1.5.4: Find a row operation and the corresponding elementry matrix that wi...
 1.5.5: In each part, an elementary matrix E and a matrix A are given. Writ...
 1.5.6: In each part, an elementary matrix E and a matrix A are given. Writ...
 1.5.7: In Exercises 78, use the following matrices
 1.5.8: In Exercises 78, use the following matrices
 1.5.9: Find an elementary matrix E that satisfies the equation. (a) (b) (c...
 1.5.10: Find an elementary matrix E that satisfies the equation. (a) (b) (c...
 1.5.11: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.12: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.13: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.14: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.15: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.16: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.17: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.18: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.19: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.20: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.21: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.22: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.23: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.24: In Exercises 924, use the inversion algorithm to find the inverse o...
 1.5.25: In Exercises 2526, find the inverse of each of the following matric...
 1.5.26: In Exercises 2526, find the inverse of each of the following matric...
 1.5.27: In Exercise 27Exercise 28, find all values of c, if any, for which ...
 1.5.28: In Exercise 27Exercise 28, find all values of c, if any, for which ...
 1.5.29: In Exercises 2932, write the given matrix as a product of elementar...
 1.5.30: In Exercises 2932, write the given matrix as a product of elementar...
 1.5.31: In Exercises 2932, write the given matrix as a product of elementar...
 1.5.32: In Exercises 2932, write the given matrix as a product of elementar...
 1.5.33: In Exercises 3336, write the inverse of the given matrix as a produ...
 1.5.34: In Exercises 3336, write the inverse of the given matrix as a produ...
 1.5.35: In Exercises 3336, write the inverse of the given matrix as a produ...
 1.5.36: In Exercises 3336, write the inverse of the given matrix as a produ...
 1.5.37: In Exercises 3738, show that the given matrices A and B are row equ...
 1.5.38: In Exercises 3738, show that the given matrices A and B are row equ...
 1.5.39: Show that if is an elementary matrix, then at least one entry in th...
 1.5.40: Show that is not invertible for any values of the entries.
 1.5.41: Prove that if A and B are matrices, then A and B are row equivalent...
 1.5.42: Prove that if A is an invertible matrix and B is row equivalent to ...
 1.5.43: Show that if B is obtained from A by performing a sequence of eleme...
 1.5.a: In parts (a)(g) determine whether the statement is true or false, a...
 1.5.b: In parts (a)(g) determine whether the statement is true or false, a...
 1.5.c: In parts (a)(g) determine whether the statement is true or false, a...
 1.5.d: In parts (a)(g) determine whether the statement is true or false, a...
 1.5.e: In parts (a)(g) determine whether the statement is true or false, a...
 1.5.f: In parts (a)(g) determine whether the statement is true or false, a...
 1.5.g: In parts (a)(g) determine whether the statement is true or false, a...
Solutions for Chapter 1.5: Elementary Matrices and a Method for Finding A1
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 1.5: Elementary Matrices and a Method for Finding A1
Get Full SolutionsElementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Since 50 problems in chapter 1.5: Elementary Matrices and a Method for Finding A1 have been answered, more than 14160 students have viewed full stepbystep solutions from this chapter. Chapter 1.5: Elementary Matrices and a Method for Finding A1 includes 50 full stepbystep solutions.

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.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.