 4.4.1: Prove that the following sets are bases for R2 by showing that they...
 4.4.2: Use Theorem 4.11 to prove that the following sets are bases forR2 (...
 4.4.3: Which of the following sets of vectors are bases for R 2? (a) {(3, ...
 4.4.4: Prove that the following sets are bases for R3 (a) {(l, 1, 1), (0, ...
 4.4.5: Which of the following sets are bases for R3? (a) {{1, 1, 2), (2, ...
 4.4.6: Explain, without performing any computation, why the following sets...
 4.4.7: Prove that the subspace of R3 generated by the vectors (1, 2, 1), ...
 4.4.8: Prove that the vector ( 1, 2, 1 ) lies in the twodimensional subs...
 4.4.9: Prove that the vector ( 2, 1, 4) lies in the twodimensional subspa...
 4.4.10: Prove that the vector ( 3, 3, 6) lies in the onedimensional subsp...
 4.4.11: Does the vector (1, 2, 1) lie in the subspace ofR3 generated by th...
 4.4.12: Find a basis for R2 that includes the vector { 1, 2).
 4.4.13: Find a basis for R3 that includes the vectors { 1, 1, 1) and (1, 0,...
 4.4.14: Find a basisforR3 that includesthe vectors (1, 0, 2) and (0, 1, 1).
 4.4.15: Determine a basis for each of the following subspaces of R3 Give th...
 4.4.16: Determine a basis for each of the following subspaces of R4 Give th...
 4.4.17: Determine a basis for each of the following vector spaces and give ...
 4.4.18: Consider the vector space M 22 of real 2 X 2 matrices. Let Vi be th...
 4.4.19: (a) Is the functionj(x) = x + 5 in the subspace spanned by g(x) = x...
 4.4.20: Are the following sets bases for the given vector spaces? (a) {J, g...
 4.4.21: Are the following sets bases for the given vector spaces? (a) {f, g...
 4.4.22: Let {Vi. v2} be a basis for a vector space V. Show that the set of ...
 4.4.23: Let {Vi. v2, v3 } be a basis for a vector space V. Show that the se...
 4.4.24: Let {vi. v2, ... , vn} be a basis for a vector space V. Let c be a ...
 4.4.25: Let V be a vector space of dimension n. Prove that no set of n  1 ...
 4.4.26: Let V be a vector space, and let W be a subspace of V. If dim(V ) =...
 4.4.27: Let u = (ui, u2) be a nonzero vector in R 2 . Prove that the set of...
 4.4.28: Let u = (ui. u2, u3 ) be a nonzero vector in R3 Prove that the set ...
 4.4.29: Let u = (ui, u2, u3) and v = (Vi, v2, v3 ) be two nonzero, linearly...
 4.4.30: Let U and V be two subspaces of a vector space W. U is said to be o...
 4.4.31: Let Vbe a vector space of dimension n. Let S = {vi. ... , v m} span...
 4.4.32: State (with a brief explanation) whether the following statements a...
 4.4.33: State (with a brief explanation) whether the following statements a...
Solutions for Chapter 4.4: Properties of Bases
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 4.4: Properties of Bases
Get Full SolutionsChapter 4.4: Properties of Bases includes 33 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Since 33 problems in chapter 4.4: Properties of Bases have been answered, more than 8756 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

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

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).

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·