×
×

Solutions for Chapter 1.5: Basis and Dimension in Rn

Full solutions for Linear Algebra with Applications | 8th Edition

ISBN: 9781449679545

Solutions for Chapter 1.5: Basis and Dimension in Rn

Solutions for Chapter 1.5
4 5 0 288 Reviews
18
4
ISBN: 9781449679545

Chapter 1.5: Basis and Dimension in Rn includes 19 full step-by-step solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Since 19 problems in chapter 1.5: Basis and Dimension in Rn have been answered, more than 9553 students have viewed full step-by-step solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545.

Key Math Terms and definitions covered in this textbook
• 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.)

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.

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Dimension of vector space

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

• 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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• 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.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Linear combination cv + d w or L C jV j.

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• 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).

• Nilpotent matrix N.

Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

• 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 subspaces.

Every v in V is orthogonal to every w in W.

• Particular solution x p.

Any solution to Ax = b; often x p has free variables = o.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

×