 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 7969 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.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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).

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

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

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

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.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.
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