 6.1: In Exercises 1 4, find (a) the image of and (b) the preimage of for...
 6.2: In Exercises 1 4, find (a) the image of and (b) the preimage of for...
 6.3: In Exercises 1 4, find (a) the image of and (b) the preimage of for...
 6.4: In Exercises 1 4, find (a) the image of and (b) the preimage of for...
 6.5: In Exercises 512, determine whether the function is a linear transf...
 6.6: In Exercises 512, determine whether the function is a linear transf...
 6.7: In Exercises 512, determine whether the function is a linear transf...
 6.8: In Exercises 512, determine whether the function is a linear transf...
 6.9: In Exercises 512, determine whether the function is a linear transf...
 6.10: In Exercises 512, determine whether the function is a linear transf...
 6.11: In Exercises 512, determine whether the function is a linear transf...
 6.12: In Exercises 512, determine whether the function is a linear transf...
 6.13: Let be a linear transformation from to such that and Find and
 6.14: Let be a linear transformation from to such that and Find
 6.15: Let be a linear transformation from to such that and Find
 6.16: Let be a linear transformation from to such that and Find
 6.17: In Exercises 1720, find the indicated power of the standard matrix ...
 6.18: In Exercises 1720, find the indicated power of the standard matrix ...
 6.19: In Exercises 1720, find the indicated power of the standard matrix ...
 6.20: In Exercises 1720, find the indicated power of the standard matrix ...
 6.21: In Exercises 2128, the linear transformation is defined by For each...
 6.22: In Exercises 2128, the linear transformation is defined by For each...
 6.23: In Exercises 2128, the linear transformation is defined by For each...
 6.24: In Exercises 2128, the linear transformation is defined by For each...
 6.25: In Exercises 2128, the linear transformation is defined by For each...
 6.26: In Exercises 2128, the linear transformation is defined by For each...
 6.27: In Exercises 2128, the linear transformation is defined by For each...
 6.28: In Exercises 2128, the linear transformation is defined by For each...
 6.29: In Exercises 2932, find a basis for (a) and
 6.30: In Exercises 2932, find a basis for (a) and
 6.31: In Exercises 2932, find a basis for (a) and
 6.32: In Exercises 2932, find a basis for (a) and
 6.33: In Exercises 3336, the linear transformation is given by Find a bas...
 6.34: In Exercises 3336, the linear transformation is given by Find a bas...
 6.35: In Exercises 3336, the linear transformation is given by Find a bas...
 6.36: In Exercises 3336, the linear transformation is given by Find a bas...
 6.37: Given and find
 6.38: Given and find
 6.39: Given and find
 6.40: Given and find
 6.41: In Exercises 4148, determine whether the transformation has an inve...
 6.42: In Exercises 4148, determine whether the transformation has an inve...
 6.43: In Exercises 4148, determine whether the transformation has an inve...
 6.44: In Exercises 4148, determine whether the transformation has an inve...
 6.45: In Exercises 4148, determine whether the transformation has an inve...
 6.46: In Exercises 4148, determine whether the transformation has an inve...
 6.47: In Exercises 4148, determine whether the transformation has an inve...
 6.48: In Exercises 4148, determine whether the transformation has an inve...
 6.49: In Exercises 49 and 50, find the standard matrices for and
 6.50: In Exercises 49 and 50, find the standard matrices for and
 6.51: Use the standard matrix for counterclockwise rotation in to rotate ...
 6.52: Rotate the triangle in Exercise 51 counterclockwise about the point...
 6.53: In Exercises 5356, determine whether the linear transformation repr...
 6.54: In Exercises 5356, determine whether the linear transformation repr...
 6.55: In Exercises 5356, determine whether the linear transformation repr...
 6.56: In Exercises 5356, determine whether the linear transformation repr...
 6.57: In Exercises 57 and 58, find by using (a) the standard matrix and (...
 6.58: In Exercises 57 and 58, find by using (a) the standard matrix and (...
 6.59: In Exercises 59 and 60, find the matrix for relative to the basis a...
 6.60: In Exercises 59 and 60, find the matrix for relative to the basis a...
 6.61: Let be represented by where (a) Find the standard matrix for (b) Le...
 6.62: Let be represented by where (a) Find the standard matrix for and sh...
 6.63: Let and be linear transformations from into Show that and are both ...
 6.64: Suppose and are similar matrices and is invertible. (a) Prove that ...
 6.65: In Exercises 65 and 66, the sum of two linear transformations and i...
 6.66: In Exercises 65 and 66, the sum of two linear transformations and i...
 6.67: Let such that (a) Prove that is linear. (b) Find the rank and nulli...
 6.68: Let and be linear transformations. (a) Prove that if and are both o...
 6.69: Let be an inner product space. For a fixed nonzero vector in let be...
 6.70: Calculus Let be a basis for a subspace W of the space of continuous...
 6.71: Writing Under what conditions are the spaces and isomorphic? Descri...
 6.72: Calculus Let be represented by Find the rank and nullity of T
 6.73: In Exercises 7378, (a) identify the transformation and (b) graphica...
 6.74: In Exercises 7378, (a) identify the transformation and (b) graphica...
 6.75: In Exercises 7378, (a) identify the transformation and (b) graphica...
 6.76: In Exercises 7378, (a) identify the transformation and (b) graphica...
 6.77: In Exercises 7378, (a) identify the transformation and (b) graphica...
 6.78: In Exercises 7378, (a) identify the transformation and (b) graphica...
 6.79: In Exercises 7982, sketch the image of the triangle with vertices a...
 6.80: In Exercises 7982, sketch the image of the triangle with vertices a...
 6.81: In Exercises 7982, sketch the image of the triangle with vertices a...
 6.82: In Exercises 7982, sketch the image of the triangle with vertices a...
 6.83: In Exercises 83 and 84, give a geometric description of the linear ...
 6.84: In Exercises 83 and 84, give a geometric description of the linear ...
 6.85: In Exercises 8588, find the matrix that will produce the indicated ...
 6.86: In Exercises 8588, find the matrix that will produce the indicated ...
 6.87: In Exercises 8588, find the matrix that will produce the indicated ...
 6.88: In Exercises 8588, find the matrix that will produce the indicated ...
 6.89: In Exercises 8992, determine the matrix that will produce the indic...
 6.90: In Exercises 8992, determine the matrix that will produce the indic...
 6.91: In Exercises 8992, determine the matrix that will produce the indic...
 6.92: In Exercises 8992, determine the matrix that will produce the indic...
 6.93: In Exercises 9396, find the image of the unit cube with vertices an...
 6.94: In Exercises 9396, find the image of the unit cube with vertices an...
 6.95: In Exercises 9396, find the image of the unit cube with vertices an...
 6.96: In Exercises 9396, find the image of the unit cube with vertices an...
 6.97: True or False? In Exercises 97100, determine whether each statement...
 6.98: True or False? In Exercises 97100, determine whether each statement...
 6.99: True or False? In Exercises 97100, determine whether each statement...
 6.100: True or False? In Exercises 97100, determine whether each statement...
Solutions for Chapter 6: Linear Transformations
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 6: Linear Transformations
Get Full SolutionsSince 100 problems in chapter 6: Linear Transformations have been answered, more than 18112 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. Chapter 6: Linear Transformations includes 100 full stepbystep solutions. 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.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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.

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 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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Solvable system Ax = b.
The right side b is in the column space of A.

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.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.