 4.2.1: For 114, determine whether the given set S of vectors is closed und...
 4.2.2: For 114, determine whether the given set S of vectors is closed und...
 4.2.3: For 114, determine whether the given set S of vectors is closed und...
 4.2.4: For 114, determine whether the given set S of vectors is closed und...
 4.2.5: For 114, determine whether the given set S of vectors is closed und...
 4.2.6: For 114, determine whether the given set S of vectors is closed und...
 4.2.7: For 114, determine whether the given set S of vectors is closed und...
 4.2.8: For 114, determine whether the given set S of vectors is closed und...
 4.2.9: For 114, determine whether the given set S of vectors is closed und...
 4.2.10: For 114, determine whether the given set S of vectors is closed und...
 4.2.11: For 114, determine whether the given set S of vectors is closed und...
 4.2.12: For 114, determine whether the given set S of vectors is closed und...
 4.2.13: For 114, determine whether the given set S of vectors is closed und...
 4.2.14: For 114, determine whether the given set S of vectors is closed und...
 4.2.15: We have defined the set R2 = {(x, y) : x, y R}, together with the a...
 4.2.16: Determine the zero vector in the vector space V = M42(R), and write...
 4.2.17: Generalize the previous exercise to find the zero vector and the ad...
 4.2.18: Determine the zero vector in the vector space V = P3(R), and write ...
 4.2.19: Generalize the previous exercise to find the zero vector and the ad...
 4.2.20: On R+, the set of positive real numbers, define the operations of a...
 4.2.21: On R2, define the operations of addition and scalar multiplication ...
 4.2.22: On R2, define the operations of addition and scalar multiplication ...
 4.2.23: On R2, define the operation of addition by (x1, y1) (x2, y2) = (x1x...
 4.2.24: On M2(R), define the operation of addition by A B = AB,and use the ...
 4.2.25: On M2(R), define the operations of addition and scalar multiplicati...
 4.2.26: For 2627, verify that the given set of objects together with the us...
 4.2.27: For 2627, verify that the given set of objects together with the us...
 4.2.28: Is C3 a real vector space? Explain.
 4.2.29: Is R3 a complex vector space? Explain.
 4.2.30: Prove part (3) of Theorem 4.2.7
 4.2.31: Prove part (6) of Theorem 4.2.7.
 4.2.32: Prove that Pn(R) is a vector space.
Solutions for Chapter 4.2: Definition of a Vector Space
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 4.2: Definition of a Vector Space
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations was written by and is associated to the ISBN: 9780321964670. Chapter 4.2: Definition of a Vector Space includes 32 full stepbystep solutions. Since 32 problems in chapter 4.2: Definition of a Vector Space have been answered, more than 19780 students have viewed full stepbystep solutions from this chapter.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

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

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).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

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).

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).