 1.8.1E: Let Find the images under
 1.8.2E: Let Define T : ?3 ??3 by T(x) = Ax. Find T(u) and T(v).
 1.8.3E: In Exercises with T defined by T(x) = Ax, find a vector × whose ima...
 1.8.4E: In Exercises with T defined by T(x) = Ax, find a vector × whose ima...
 1.8.5E: In Exercises 3–6, with T defined by T(x) = Ax, find a vector x whos...
 1.8.6E: In Exercises with T defined by T(x) = Ax, find a vector × whose ima...
 1.8.7E: Let A be a 6 × 5 matrix. What must a and b be in order to define
 1.8.8E: How many rows and columns must a matrix A have in order to define a...
 1.8.9E: Find all × in ?4 that are mapped into the zero vector by the transf...
 1.8.10E: Find all × in ?4 that are mapped into the zero vector by the transf...
 1.8.11E: Let and let A be the matrix in Exercise 9. Is b in the range of the...
 1.8.12E: Let and let A be the matrix in Exercise 10. Is b in the range of th...
 1.8.13E: In Exercises 13–16, use a rectangular coordinate system to plot and...
 1.8.14E: Use a rectangular coordinate system to plot and their images under ...
 1.8.15E: Use a rectangular coordinate system to plot and their images under ...
 1.8.16E: In Exercises 13–16, use a rectangular coordinate system to plot and...
 1.8.17E: Let T : ?2 ? ?2 be a linear transformation that maps and maps . Use...
 1.8.18E: The figure shows vectors u, v, and w, along with the images T (u) a...
 1.8.19E: and let T: ?2 ? ?2 be a linear transformation that maps e1 into y1 ...
 1.8.20E: Let , and let T : ?2 ? ?2be a linear transformation that maps × int...
 1.8.21E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 1.8.22E: In Exercises mark each statement True or False. Justify each answer...
 1.8.23E: Let T : ?2 ? ?2 be the linear transformation that reflects each poi...
 1.8.24E: Suppose vectors be a linear transformation. Suppose . Show that T i...
 1.8.25E: Given v ? 0 and p in ?n, the line through p in the direction of v h...
 1.8.26E: Let u and v be linearly independent vectors in ?3, and let P be the...
 1.8.27E: a. Show that the line through vectors p and q in ?n may be written ...
 1.8.28E: Let u and v be vectors in ?n. It can be shown that the set P of all...
 1.8.29E: Define a. Show that f is a linear transformation when b = 0.b. Find...
 1.8.30E: An affine transformation T: ?n ? ?m has the form T (x) = Ax + b, wi...
 1.8.31E: Let T : ?n ? ?m be a linear transformation, and let {v1, v2, v3} be...
 1.8.32E: In Exercises, column vectors are written as rows, such as × = (x1, ...
 1.8.33E: Show that the transformation T defined by T(x1,x2) = (2x1  3x2, x1...
 1.8.34E: In Exercises 32–36, column vectors are written as rows, such as x =...
 1.8.35E: In Exercises 32–36, column vectors are written as rows, such as x =...
 1.8.36E: In Exercises 32–36, column vectors are written as rows, such as x =...
 1.8.37E: [M] In Exercises the given matrix determines a linear transformatio...
 1.8.38E: [M] In Exercises the given matrix determines a linear transformatio...
 1.8.39E: [M] Let b = and let A be the matrix in Exercise. Is b in the range ...
 1.8.40E: [M] Let b = and let A be the matrix in Exercise.Is b in the range o...
Solutions for Chapter 1.8: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 1.8
Get Full SolutionsChapter 1.8 includes 40 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 40 problems in chapter 1.8 have been answered, more than 39848 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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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).

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

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