 4.4.1: In each case, a linear transformation T : M22 M22 is defined. Give ...
 4.4.2: Let V C (R) be the given subspace. Let D: V V be the differentiatio...
 4.4.3: Verify the details of Example 12.
 4.4.4: Use the changeofbasis formula to find the matrix for the linear t...
 4.4.5: Define T : P3 P3 by T (f )(t) = 2f (t) + (1 t)f (t ). a. Show that ...
 4.4.6: Consider the differentiation operator D: C1(R) C0(R) (or Pk Pk1, if...
 4.4.7: Define M: P P by M(f )(t) = tf (t), and let D: P P be the different...
 4.4.8: Let V and W be vector spaces, and let T : V W be a linear transform...
 4.4.9: a. Consider the identity transformation Id : Rn Rn. Using the basis...
 4.4.10: Give a proof of Theorem 4.2 modeled on the proof of Proposition 3.2.
 4.4.11: Let V andW be vector spaces (not necessarily finitedimensional), a...
 4.4.12: Suppose V = {v1, . . . , vn} is an ordered basis for V ,W = {w1, . ...
 4.4.13: Prove that if a linear transformation T : V W is an isomorphism, th...
 4.4.14: Decide whether each of the following functions T is a linear transf...
 4.4.15: Let I = [0, 1], and let M: C0(I) C0(I) be given by M(f )(t) = tf (t...
 4.4.16: Let V be a finitedimensional vector space, letW be a vector space,...
 4.4.17: Suppose T : V W is an isomorphism and dim V = n. Prove that dimW = n.
 4.4.18: a. Suppose T : V W is a linear transformation. Suppose {v1, . . . ,...
 4.4.19: Let V and W be subspaces of Rn with V W = {0}. Let S = projV and T ...
 4.4.20: Suppose V is a vector space and T : V V is a linear transformation....
 4.4.21: Let V be a vector space. a. Let V denote the set of all linear tran...
 4.4.22: Let t1, . . . , tk+1 be distinct real numbers. Define a linear tran...
 4.4.23: Suppose T : Rn Rn has the following properties: (i) T (0) = 0; (ii)...
 4.4.24: (See the discussion on p. 167 and Exercise 3.4.25.) Let A be an n n...
Solutions for Chapter 4.4: Linear Transformations on Abstract Vector Spaces
Full solutions for Linear Algebra: A Geometric Approach  2nd Edition
ISBN: 9781429215213
Solutions for Chapter 4.4: Linear Transformations on Abstract Vector Spaces
Get Full SolutionsSince 24 problems in chapter 4.4: Linear Transformations on Abstract Vector Spaces have been answered, more than 4413 students have viewed full stepbystep solutions from this chapter. Linear Algebra: A Geometric Approach was written by and is associated to the ISBN: 9781429215213. This textbook survival guide was created for the textbook: Linear Algebra: A Geometric Approach, edition: 2. Chapter 4.4: Linear Transformations on Abstract Vector Spaces includes 24 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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).

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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).

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.