 1.9.1E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.2E: In Exercises, assume that T is a linear transformation. Find the st...
 1.9.3E: In Exercises, assume that T is a linear transformation. Find the st...
 1.9.4E: In Exercises, assume that T is a linear transformation. Find the st...
 1.9.5E: In Exercises, assume that T is a linear transformation. Find the st...
 1.9.6E: In Exercises, assume that T is a linear transformation. Find the st...
 1.9.7E: In Exercises 1–10, assume that T is a linear transformation. Find t...
 1.9.8E: 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, fill in the missing entries of the matrix, assuming t...
 1.9.16E: In Exercises, fill in the missing entries of the matrix, assuming t...
 1.9.17E: In Exercises, show that T is a linear transformation by finding a m...
 1.9.18E: In Exercises, show that T is a linear transformation by finding a m...
 1.9.19E: In Exercises 17–20, show that T is a linear transformation by findi...
 1.9.20E: In Exercises, show that T is a linear transformation by finding a m...
 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 such that T(x1,x2) = (...
 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: Why is the question “Is the linear transformation T onto?” an exist...
 1.9.35E: If a linear transformation can you give a relation between m and n?...
 1.9.36E: be linear transformations. Show that the mapping is a linear transf...
 1.9.37E: [M] In Exercises, let T be the linear transformation whose standard...
 1.9.38E: [M] In Exercises, let T be the linear transformation whose standard...
 1.9.39E: [M] In Exercises 37–40, let T be the linear transformation whose st...
 1.9.40E: [M] In Exercises, let T be the linear transformation whose standard...
Solutions for Chapter 1.9: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 1.9
Get Full SolutionsChapter 1.9 includes 40 full stepbystep solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 40 problems in chapter 1.9 have been answered, more than 43727 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. This expansive textbook survival guide covers the following chapters and their solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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 •

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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 ].

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.