 3.1.1E: Sketch the geometric vector (with initial point at the origin) corr...
 3.1.2E: Sketch the geometric vector (with initial point at the origin) corr...
 3.1.3E: Sketch the geometric vector (with initial point at the origin) corr...
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 3.1.6E: Sketch the geometric vector (with initial point at the origin) corr...
 3.1.7E: Sketch the geometric vector (with initial point at the origin) corr...
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 3.1.9E: Sketch the geometric vector (with initial point at the origin) corr...
 3.1.10E: Sketch the geometric vector (with initial point at the origin) corr...
 3.1.11E: Sketch the geometric vector (with initial point at the origin) corr...
 3.1.12E: Interpret the subset W of R2 geometrically by sketching a graph for W.
 3.1.13E: Interpret the subset W of R2 geometrically by sketching a graph for W.
 3.1.14E: Interpret the subset W of R2 geometrically by sketching a graph for W.
 3.1.15E: Interpret the subset W of R2 geometrically by sketching a graph for W.
 3.1.16E: Interpret the subset W of R2 geometrically by sketching a graph for W.
 3.1.17E: Interpret the subset W of R2 geometrically by sketching a graph for W.
 3.1.18E: Interpret the subset W of R3 geometrically by sketching a graph for W.
 3.1.19E: Interpret the subset W of R3 geometrically by sketching a graph for W.
 3.1.20E: Interpret the subset W of R3 geometrically by sketching a graph for W.
 3.1.21E: Interpret the subset W of R3 geometrically by sketching a graph for W.
 3.1.22E: Give a settheoretic description of the given points as a subset W ...
 3.1.23E: Give a settheoretic description of the given points as a subset W ...
 3.1.24E: Give a settheoretic description of the given points as a subset W ...
 3.1.25E: Give a settheoretic description of the given points as a subset W ...
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 3.1.30E: Give a settheoretic description of the given points as a subset W ...
Solutions for Chapter 3.1: Introduction to Linear Algebra 5th Edition
Full solutions for Introduction to Linear Algebra  5th Edition
ISBN: 9780201658590
Solutions for Chapter 3.1
Get Full SolutionsThis textbook survival guide was created for the textbook: Introduction to Linear Algebra , edition: 5. Since 30 problems in chapter 3.1 have been answered, more than 7473 students have viewed full stepbystep solutions from this chapter. Chapter 3.1 includes 30 full stepbystep solutions. Introduction 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.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.)

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.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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 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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.