 4.3.1: Are the polynomials /(f) = 1 I 3f + /2, g(t) = 9 + 9f 4 4f2, and...
 4.3.2: Are the matrices1 1 1 2 2 3 1 4 1 1 3 4 5 7 6 8linearly independent?
 4.3.3: Do the polynomials /(f) = 1 + 2/ + 9f2 + f3, g(/) = 1 + It + 7f3, /...
 4.3.4: Consider the polynomials /(f) = t + 1 and g(t) = (f + 2)(f + k), wh...
 4.3.5: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.6: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.7: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.8: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.9: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.10: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.11: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.12: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.13: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.14: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.15: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.16: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.17: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.18: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.19: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.20: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.21: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.22: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.23: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.24: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.25: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.26: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.27: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.28: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.29: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.30: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.31: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.32: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.33: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.34: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.35: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.36: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.37: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.38: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.39: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.40: In Exercises 5 through 40, find the matrix of the given linear tran...
 4.3.41: a. Find the change of basis matrix S from the basis 93 considered i...
 4.3.42: considered in Exercise 7 to the standard basis 21 of 1 5' U2x2 cons...
 4.3.43: a. Find the change of basis matrix S from the basis 93 considered i...
 4.3.44: a. Find the change of basis matrix S from the basis 93 considered i...
 4.3.45: a. Find the change of basis matrix S from the basis 33 considered i...
 4.3.46: a. Find the change of basis matrix S from the basis 93 considered i...
 4.3.47: a. Find the change of basis matrix S from the basis 93 considered i...
 4.3.48: In Exercises 48 through 53, let V be the space spanned by the two f...
 4.3.49: In Exercises 48 through 53, let V be the space spanned by the two f...
 4.3.50: In Exercises 48 through 53, let V be the space spanned by the two f...
 4.3.51: In Exercises 48 through 53, let V be the space spanned by the two f...
 4.3.52: In Exercises 48 through 53, let V be the space spanned by the two f...
 4.3.53: In Exercises 48 through 53, let V be the space spanned by the two f...
 4.3.54: In Exercises 54 through 58, let V be the plane with equation jci + ...
 4.3.55: In Exercises 54 through 58, let V be the plane with equation jci + ...
 4.3.56: In Exercises 54 through 58, let V be the plane with equation jci + ...
 4.3.57: In Exercises 54 through 58, let V be the plane with equation jci + ...
 4.3.58: In Exercises 54 through 58, let V be the plane with equation jci + ...
 4.3.59: Consider a linear transformation T from V to V with ker(D = {0}. If...
 4.3.60: In the plane V defined by the equation 2x\ +X2  2jc3 = 0, consider...
 4.3.61: In R2, consider the basesand1 = (21, a2) =a* Find the change of bas...
 4.3.62: In the plane V defined by the equation x i 2x2 + 2x3 = 0, consider ...
 4.3.63: In the plane V defined by the equation x\ + 3*2 2x3 = 0, consider t...
 4.3.64: Let V be the space of all upper triangular 2x2 matrices. Consider t...
 4.3.65: Let V be the subspace of R2x2 spanned by the matrices where b ^ 0.a...
 4.3.66: Let V be the linear space of all functions in two variables of the ...
 4.3.67: Let V be the linear space of all functions of the form/(/) = ci cos...
 4.3.68: Consider the linear space V of all infinite sequences of real numbe...
 4.3.69: Consider a basis / 1, ..., f n of Pn \ . Let a \,..., an be distin...
 4.3.70: Consider two finite dimensional linear spaces V and W. If V and W a...
 4.3.71: Let a\,..., an be distinct real numbers. Show that there exist weig...
 4.3.72: Find the weights w\,w2,wi in Exercise 71 for a\ ~ 1 ,a2 = 0,03 = 1....
Solutions for Chapter 4.3: Th e Matrix of a Linear Transformation
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 4.3: Th e Matrix of a Linear Transformation
Get Full SolutionsChapter 4.3: Th e Matrix of a Linear Transformation includes 72 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Since 72 problems in chapter 4.3: Th e Matrix of a Linear Transformation have been answered, more than 15447 students have viewed full stepbystep solutions from this chapter.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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!).

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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}.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.

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