 2.6.1: 114 compute the factorization A = LV (and also A = LDV).
 2.6.2: 114 compute the factorization A = LV (and also A = LDV).
 2.6.3: 114 compute the factorization A = LV (and also A = LDV).
 2.6.4: 114 compute the factorization A = LV (and also A = LDV).
 2.6.5: 114 compute the factorization A = LV (and also A = LDV).
 2.6.6: 114 compute the factorization A = LV (and also A = LDV).
 2.6.7: 114 compute the factorization A = LV (and also A = LDV).
 2.6.8: 114 compute the factorization A = LV (and also A = LDV).
 2.6.9: 114 compute the factorization A = LV (and also A = LDV).
 2.6.10: 114 compute the factorization A = LV (and also A = LDV).
 2.6.11: 114 compute the factorization A = LV (and also A = LDV).
 2.6.12: 114 compute the factorization A = LV (and also A = LDV).
 2.6.13: 114 compute the factorization A = LV (and also A = LDV).
 2.6.14: 114 compute the factorization A = LV (and also A = LDV).
 2.6.15: 1516 use Land U (without needing A) to solve Ax = b.
 2.6.16: 1516 use Land U (without needing A) to solve Ax = b.
 2.6.17: (a) When you apply the usual elimination steps to L, what matrix do...
 2.6.18: If A = LDU and also A = Ll Dl U1 with all factors invertible, then ...
 2.6.19: Tridiagonal matrices have zero entries except on the main diagonal ...
 2.6.20: When T is tridiagonal, its Land U factors have only two nonzero dia...
 2.6.21: If A and B have nonzeros in the positions marked by x, which zeros ...
 2.6.22: Suppose you eliminate upwards (almost unheard ot). Use the last row...
 2.6.23: Easy but important. If A has pivots 5, 9, 3 with no row exchanges, ...
 2.6.24: Which invertible matrices allow A = LV (elimination without row exc...
 2.6.25: For the 6 by 6 second difference constantdiagonal matrix K, put th...
 2.6.26: f you print K 1 , ;it doesn't look so good. But if you print 7 K ...
Solutions for Chapter 2.6: Elimination = Factorization: A = L U
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 2.6: Elimination = Factorization: A = L U
Get Full SolutionsChapter 2.6: Elimination = Factorization: A = L U includes 26 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Since 26 problems in chapter 2.6: Elimination = Factorization: A = L U have been answered, more than 8145 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714.

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

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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].

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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 •

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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