 6.3.1: In Exercises 110, find the standard matrix for the linear transform...
 6.3.2: In Exercises 110, find the standard matrix for the linear transform...
 6.3.3: In Exercises 110, find the standard matrix for the linear transform...
 6.3.4: In Exercises 110, find the standard matrix for the linear transform...
 6.3.5: In Exercises 110, find the standard matrix for the linear transform...
 6.3.6: In Exercises 110, find the standard matrix for the linear transform...
 6.3.7: In Exercises 110, find the standard matrix for the linear transform...
 6.3.8: In Exercises 110, find the standard matrix for the linear transform...
 6.3.9: In Exercises 110, find the standard matrix for the linear transform...
 6.3.10: In Exercises 110, find the standard matrix for the linear transform...
 6.3.11: In Exercises 1116, use the standard matrix for the linear transform...
 6.3.12: In Exercises 1116, use the standard matrix for the linear transform...
 6.3.13: In Exercises 1116, use the standard matrix for the linear transform...
 6.3.14: In Exercises 1116, use the standard matrix for the linear transform...
 6.3.15: In Exercises 1116, use the standard matrix for the linear transform...
 6.3.16: In Exercises 1116, use the standard matrix for the linear transform...
 6.3.17: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.18: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.19: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.20: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.21: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.22: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.23: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.24: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.25: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.26: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.27: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.28: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.29: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.30: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.31: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.32: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.33: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.34: In Exercises 1734, (a) find the standard matrix for the linear tran...
 6.3.35: In Exercises 3538, (a) find the standard matrix for the linear tran...
 6.3.36: In Exercises 3538, (a) find the standard matrix for the linear tran...
 6.3.37: In Exercises 3538, (a) find the standard matrix for the linear tran...
 6.3.38: In Exercises 3538, (a) find the standard matrix for the linear tran...
 6.3.39: In Exercises 3944, find the standard matrices for and
 6.3.40: In Exercises 3944, find the standard matrices for and
 6.3.41: In Exercises 3944, find the standard matrices for and
 6.3.42: In Exercises 3944, find the standard matrices for and
 6.3.43: In Exercises 3944, find the standard matrices for and
 6.3.44: In Exercises 3944, find the standard matrices for and
 6.3.45: In Exercises 4556, determine whether the linear transformation is i...
 6.3.46: In Exercises 4556, determine whether the linear transformation is i...
 6.3.47: In Exercises 4556, determine whether the linear transformation is i...
 6.3.48: In Exercises 4556, determine whether the linear transformation is i...
 6.3.49: In Exercises 4556, determine whether the linear transformation is i...
 6.3.50: In Exercises 4556, determine whether the linear transformation is i...
 6.3.51: In Exercises 4556, determine whether the linear transformation is i...
 6.3.52: In Exercises 4556, determine whether the linear transformation is i...
 6.3.53: In Exercises 4556, determine whether the linear transformation is i...
 6.3.54: In Exercises 4556, determine whether the linear transformation is i...
 6.3.55: In Exercises 4556, determine whether the linear transformation is i...
 6.3.56: In Exercises 4556, determine whether the linear transformation is i...
 6.3.57: In Exercises 5764, find by using (a) the standard matrix and (b) th...
 6.3.58: In Exercises 5764, find by using (a) the standard matrix and (b) th...
 6.3.59: In Exercises 5764, find by using (a) the standard matrix and (b) th...
 6.3.60: In Exercises 5764, find by using (a) the standard matrix and (b) th...
 6.3.61: In Exercises 5764, find by using (a) the standard matrix and (b) th...
 6.3.62: In Exercises 5764, find by using (a) the standard matrix and (b) th...
 6.3.63: In Exercises 5764, find by using (a) the standard matrix and (b) th...
 6.3.64: In Exercises 5764, find by using (a) the standard matrix and (b) th...
 6.3.65: Let be given by Find the matrix for relative to the bases and
 6.3.66: Let be given by Find the matrix for relative to the bases and
 6.3.67: Calculus Let be a basis of a subspace of the space of continuous fu...
 6.3.68: Calculus Repeat Exercise 67 for
 6.3.69: Calculus Use the matrix from Exercise 67 to evaluate
 6.3.70: Calculus Use the matrix from Exercise 68 to evaluate
 6.3.71: Calculus Let be a basis for and let be the linear transformation re...
 6.3.72: (a) If is a linear transformation to the matrix is called the stand...
 6.3.73: (a) The composition T of linear transformations and represented by ...
 6.3.74: Let be a linear transformation such that for in Find the standard m...
 6.3.75: Let be represented by Find the matrix for relative to the standard ...
 6.3.76: Show that the linear transformation given in Exercise 75 is an isom...
 6.3.77: Guided Proof Let and be onetoone linear transformations. Prove th...
 6.3.78: Prove Theorem 6.12.
 6.3.79: Writing Is it always preferable to use the standard basis for Discu...
 6.3.80: Writing Look back at Theorem 4.19 and rephrase it in terms of what ...
Solutions for Chapter 6.3: Matrices for Linear Transformations
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 6.3: Matrices for Linear Transformations
Get Full SolutionsSince 80 problems in chapter 6.3: Matrices for Linear Transformations have been answered, more than 43253 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.3: Matrices for Linear Transformations includes 80 full stepbystep 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.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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

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