 6.5.1: Let be a reflection in the axis. Find the image of each vector. (a...
 6.5.2: Let be a reflection in the axis. Find the image of each vector. (a...
 6.5.3: Let be a reflection in the line Find the image of each vector. (a) ...
 6.5.4: Let be a reflection in the line Find the image of each vector. (a) ...
 6.5.5: Let and (a) Determine for any (b) Give a geometric description of T.
 6.5.6: Let and (a) Determine for any (b) Give a geometric description of T
 6.5.7: In Exercises 714, (a) identify the transformation and (b) graphical...
 6.5.8: In Exercises 714, (a) identify the transformation and (b) graphical...
 6.5.9: In Exercises 714, (a) identify the transformation and (b) graphical...
 6.5.10: In Exercises 714, (a) identify the transformation and (b) graphical...
 6.5.11: In Exercises 714, (a) identify the transformation and (b) graphical...
 6.5.12: In Exercises 714, (a) identify the transformation and (b) graphical...
 6.5.13: In Exercises 714, (a) identify the transformation and (b) graphical...
 6.5.14: In Exercises 714, (a) identify the transformation and (b) graphical...
 6.5.15: In Exercises 1522, find all fixed points of the linear transformati...
 6.5.16: In Exercises 1522, find all fixed points of the linear transformati...
 6.5.17: In Exercises 1522, find all fixed points of the linear transformati...
 6.5.18: In Exercises 1522, find all fixed points of the linear transformati...
 6.5.19: In Exercises 1522, find all fixed points of the linear transformati...
 6.5.20: In Exercises 1522, find all fixed points of the linear transformati...
 6.5.21: In Exercises 1522, find all fixed points of the linear transformati...
 6.5.22: In Exercises 1522, find all fixed points of the linear transformati...
 6.5.23: In Exercises 2328, sketch the image of the unit square with vertice...
 6.5.24: In Exercises 2328, sketch the image of the unit square with vertice...
 6.5.25: In Exercises 2328, sketch the image of the unit square with vertice...
 6.5.26: In Exercises 2328, sketch the image of the unit square with vertice...
 6.5.27: In Exercises 2328, sketch the image of the unit square with vertice...
 6.5.28: In Exercises 2328, sketch the image of the unit square with vertice...
 6.5.29: In Exercises 2934, sketch the image of the rectangle with vertices ...
 6.5.30: In Exercises 2934, sketch the image of the rectangle with vertices ...
 6.5.31: In Exercises 2934, sketch the image of the rectangle with vertices ...
 6.5.32: In Exercises 2934, sketch the image of the rectangle with vertices ...
 6.5.33: In Exercises 2934, sketch the image of the rectangle with vertices ...
 6.5.34: In Exercises 2934, sketch the image of the rectangle with vertices ...
 6.5.35: In Exercises 3538, sketch each of the images with the given vertice...
 6.5.36: In Exercises 3538, sketch each of the images with the given vertice...
 6.5.37: In Exercises 3538, sketch each of the images with the given vertice...
 6.5.38: In Exercises 3538, sketch each of the images with the given vertice...
 6.5.39: The linear transformation defined by a diagonal matrix with positiv...
 6.5.40: Repeat Exercise 39 for the linear transformation defined by
 6.5.41: In Exercises 4146, give a geometric description of the linear trans...
 6.5.42: In Exercises 4146, give a geometric description of the linear trans...
 6.5.43: In Exercises 4146, give a geometric description of the linear trans...
 6.5.44: In Exercises 4146, give a geometric description of the linear trans...
 6.5.45: In Exercises 4146, give a geometric description of the linear trans...
 6.5.46: In Exercises 4146, give a geometric description of the linear trans...
 6.5.47: In Exercises 47 and 48, give a geometric description of the linear ...
 6.5.48: In Exercises 47 and 48, give a geometric description of the linear ...
 6.5.49: In Exercises 4952, find the matrix that will produce the indicated ...
 6.5.50: In Exercises 4952, find the matrix that will produce the indicated ...
 6.5.51: In Exercises 4952, find the matrix that will produce the indicated ...
 6.5.52: In Exercises 4952, find the matrix that will produce the indicated ...
 6.5.53: In Exercises 5356, find the image of the vector for the indicated r...
 6.5.54: In Exercises 5356, find the image of the vector for the indicated r...
 6.5.55: In Exercises 5356, find the image of the vector for the indicated r...
 6.5.56: In Exercises 5356, find the image of the vector for the indicated r...
 6.5.57: In Exercises 5762, determine which single counterclockwise rotation...
 6.5.58: In Exercises 5762, determine which single counterclockwise rotation...
 6.5.59: In Exercises 5762, determine which single counterclockwise rotation...
 6.5.60: In Exercises 5762, determine which single counterclockwise rotation...
 6.5.61: In Exercises 5762, determine which single counterclockwise rotation...
 6.5.62: In Exercises 5762, determine which single counterclockwise rotation...
 6.5.63: In Exercises 6366, determine the matrix that will produce the indic...
 6.5.64: In Exercises 6366, determine the matrix that will produce the indic...
 6.5.65: In Exercises 6366, determine the matrix that will produce the indic...
 6.5.66: In Exercises 6366, determine the matrix that will produce the indic...
Solutions for Chapter 6.5: Applications of Linear Transformations
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 6.5: Applications of Linear Transformations
Get Full SolutionsSince 66 problems in chapter 6.5: Applications of Linear Transformations have been answered, more than 19611 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. Chapter 6.5: Applications of Linear Transformations includes 66 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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