 1.5.1: Give the standard basis for R2 (a) Show that the vectors in the bas...
 1.5.2: Give the standard basis for R4 (a) Show that the vectors in the bas...
 1.5.3: Consider the sets of vectors of the following form. Prove that they...
 1.5.4: Consider the sets of vectors of the following form. Determine wheth...
 1.5.5: Consider the sets of vectors of the following form. Determine wheth...
 1.5.6: State with a brief explanation whether the following statements are...
 1.5.7: State with a brief explanation whether the following statements are...
 1.5.8: (a) Show that the vectors (1, 0), (0, 1) span R2 and are also linea...
 1.5.9: a) Show that the vectors (1, 0), (0, 1) span R2 and are also linear...
 1.5.10: (a) Show that the vectors (1, 0, 0), (0, 1, 0), (0, 0, 1) span R3 a...
 1.5.11: (a) Show by means of an example that it is possible to have a set o...
 1.5.12: In Exercises 1215 consider the homogeneous systems of linear equat...
 1.5.13: In Exercises 1215 consider the homogeneous systems of linear equat...
 1.5.14: In Exercises 1215 consider the homogeneous systems of linear equat...
 1.5.15: In Exercises 1215 consider the homogeneous systems of linear equat...
 1.5.16: In Exercises 1618 consider the homogeneous systems of linear equat...
 1.5.17: In Exercises 1618 consider the homogeneous systems of linear equat...
 1.5.18: In Exercises 1618 consider the homogeneous systems of linear equat...
 1.5.19: Determine whether the following sets of vectors are linearly depend...
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
Get Full SolutionsChapter 1.5: Basis and Dimension in Rn includes 19 full stepbystep 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 stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545.

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.)

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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 IIA1 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.
Vector addition and scalar multiplication.

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 = Q1 = 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.