 4.2.1: Find out which of the transformations in Exercises 1 through 50 are...
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 4.2.51: Find kernel and nullity of the transformation in Exercise 13.
 4.2.52: Find kernel and nullity of the transformation in Exercise 6.
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 4.2.55: Find image and kernel of the transformation in Exercise 33.
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 4.2.57: Find kernel and nullity of the transformation in Exercise 39.
 4.2.58: Find image and kernel of the transformation in Exercise 34.
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 4.2.61: Find image and kernel of the transformation in Exercise 45.
 4.2.62: Find image and kernel of the transformation in Exercise 48.
 4.2.63: Define an isomorphism from P3 to R3, if you can.
 4.2.64: Define an isomorphism from P3 to R2x2, if you can.
 4.2.65: We will define a transformation T from R',xm to F(Rn\ R n)\ recall ...
 4.2.66: Find the kernel and nullity of the linear transformation T(f) = f ...
 4.2.67: For which constants k is the linear transformationT(M) =2 3 0 4M  ...
 4.2.68: For which constants k is the linear transformation= ][Jan isomorph...
 4.2.69: If matrix A is similar to #, is T{M) = AM MB an isomorphism from R2...
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 4.2.71: Does there exist a polynomial f(t) of degree < 4 such that /(2) = 3...
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 4.2.73: In Exercises 72 through 74, let Zn be the set of all polynomials of...
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 4.2.75: Show that if 0 is the neutral element of a linear space then 0 + 0 ...
 4.2.76: Show that if 7 is a linear transformation from V to W, then T(0y) =...
 4.2.77: If 7 is a linear transformation from V to W and L is a linear trans...
 4.2.78: Let R+ be the set of positive real numbers. On R+ we define the exo...
 4.2.79: Is it possible to define exotic operations on R2, so that dim(R2) = 1?
 4.2.80: Let X be the set of all students in your linear algebra class. Can ...
 4.2.81: In this exercise, we will outline a proof of the ranknullity theor...
 4.2.82: Prove the following variant of the ranknullity theorem: If T is a ...
 4.2.83: Consider linear transformations T from V to W and L from W to U. If...
 4.2.84: Consider linear transformations 7 from V to W and L from W to U. If...
Solutions for Chapter 4.2: Linear Transformations and Isomorphisms
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 4.2: Linear Transformations and Isomorphisms
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 84 problems in chapter 4.2: Linear Transformations and Isomorphisms have been answered, more than 15418 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Chapter 4.2: Linear Transformations and Isomorphisms includes 84 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

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).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

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

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.

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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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