 3.2.1E: W is a subset of R2 consisting of vectors of the form To determine ...
 3.2.2E: W is a subset of R2 consisting of vectors of the form To determine ...
 3.2.3E: W is a subset of R2 consisting of vectors of the form To determine ...
 3.2.4E: W is a subset of R2 consisting of vectors of the form To determine ...
 3.2.5E: W is a subset of R2 consisting of vectors of the form To determine ...
 3.2.6E: W is a subset of R2 consisting of vectors of the form To determine ...
 3.2.7E: W is a subset of R2 consisting of vectors of the form To determine ...
 3.2.8E: W is a subset of R2 consisting of vectors of the form To determine ...
 3.2.9E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.10E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.11E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.12E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.13E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.14E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.15E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.16E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.17E: W is a subset of R3 consisting of vectors of the form Determine whe...
 3.2.18E: Let abe a fixed vector in R3, and define W to be the subset of R3 g...
 3.2.19E: Let abe a fixed vector in R3, and define W to be the subset of R3 g...
 3.2.20E: Let abe a fixed vector in R3, and define W to be the subset of R3 g...
 3.2.21E: Let a and b be fixed vectors in R3, and let W be the subset of R3 d...
 3.2.22E: Let a and b be fixed vectors in R3, and let W be the subset of R3 d...
 3.2.23E: Let a and b be fixed vectors in R3, and let W be the subset of R3 d...
 3.2.24E: Let a and b be fixed vectors in R3, and let W be the subset of R3 d...
 3.2.25E: Let a and b be fixed vectors in R3, and let W be the subset of R3 d...
 3.2.26E: (Reference Theorem 1)
 3.2.27E: (Reference Theorem 1)
 3.2.28E: (Reference Theorem 1)
 3.2.29E: In R3, a fine through the origin is the set of all points in R3 who...
 3.2.30E: If U and V are subsets of Rn, then the set U + V is defined byU+ V ...
 3.2.31E: Let U and V be subspaces of Rn. Prove that the intersection, U ? V,...
 3.2.32E: Let U and V be the subspaces of R3 defined byU = {x: aT x = 0} and ...
 3.2.33E: Let U and V be subspaces of .a) Show that the union, U ? V, satisfi...
 3.2.34E: Let W be a nonempty subset of Rn that satisfies conditions (s2) and...
Solutions for Chapter 3.2: Introduction to Linear Algebra 5th Edition
Full solutions for Introduction to Linear Algebra  5th Edition
ISBN: 9780201658590
Solutions for Chapter 3.2
Get Full SolutionsIntroduction to Linear Algebra was written by and is associated to the ISBN: 9780201658590. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra , edition: 5. Since 34 problems in chapter 3.2 have been answered, more than 7142 students have viewed full stepbystep solutions from this chapter. Chapter 3.2 includes 34 full stepbystep solutions.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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].

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

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi 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·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.