 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...
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 6.3.9: In Exercises 110, find the standard matrix for the linear transform...
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 6.3.11: In Exercises 1116, use the standard matrix for the linear transform...
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 6.3.17: In Exercises 1734, (a) find the standard matrix for the linear tran...
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 6.3.35: In Exercises 3538, (a) find the standard matrix for the linear tran...
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 6.3.39: In Exercises 3944, find the standard matrices for and
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 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...
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 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 15117 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.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

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.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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.

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