 1.9.1E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.2E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.3E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.4E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.5E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.6E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.7E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.9E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.10E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.11E: A linear transformation T : ?2 ? ?2 first reflects points through t...
 1.9.12E: Show that the transformation in Exercise 10 is merely a rotation ab...
 1.9.13E: Let T : ?2 ? ?2 be the linear transformation such that T (e1) and T...
 1.9.14E: Let T : ?2 ? ?2 be a linear transformation with standard matrix A =...
 1.9.15E: In Exercises 15 and 16, fill in the missing entries of the matrix, ...
 1.9.16E: In Exercises 15 and 16, fill in the missing entries of the matrix, ...
 1.9.17E: In Exercises 17–20, show that T is a linear transformation by findi...
 1.9.18E: In Exercises 17–20, show that T is a linear transformation by findi...
 1.9.19E: In Exercises 17–20, show that T is a linear transformation by findi...
 1.9.20E: In Exercises 17–20, show that T is a linear transformation by findi...
 1.9.21E: Let T : ?2 ? ?2 be a linear transformation such that . Find x such ...
 1.9.22E: Let T : ?2 ? ?3 be a linear transformation with Find x such that
 1.9.23E: In Exercises 23 and 24, mark each statement True or False. Justify ...
 1.9.24E: In Exercises 23 and 24, mark each statement True or False. Justify ...
 1.9.25E: In Exercises 25–28, determine if the specified linear transformatio...
 1.9.26E: In Exercises 25–28, determine if the specified linear transformatio...
 1.9.27E: In Exercises 25–28, determine if the specified linear transformatio...
 1.9.28E: In Exercises 25–28, determine if the specified linear transformatio...
 1.9.29E: In Exercises 29 and 30, describe the possible echelon forms of the ...
 1.9.30E: In Exercises 29 and 30, describe the possible echelon forms of the ...
 1.9.31E: Let T : ?n? ?m be a linear transformation, with A its standard matr...
 1.9.32E: Let T : ?n? ?m be a linear transformation, with A its standard matr...
 1.9.33E: Verify the uniqueness of A in Theorem 10. Let T : ?n? ?m be a linea...
 1.9.34E: be linear transformations. Show that the mapping is a linear transf...
 1.9.35E: If a linear transformation can you give a relation between m and n?...
 1.9.36E: Why is the question “Is the linear transformation T onto?” an exist...
 1.9.37E: [M] In Exercises 37–40, let T be the linear transformation whose st...
 1.9.38E: [M] In Exercises 37–40, let T be the linear transformation whose st...
 1.9.39E: [M] In Exercises 37–40, let T be the linear transformation whose st...
 1.9.40E: [M] In Exercises 37–40, let T be the linear transformation whose st...
Solutions for Chapter 1.9: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 1.9
Get Full SolutionsChapter 1.9 includes 39 full stepbystep solutions. Since 39 problems in chapter 1.9 have been answered, more than 32452 students have viewed full stepbystep solutions from this chapter. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

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

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.

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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