 6.4.1: Let T1 : R2 R2 and T2 : R2 R be the linear transformations with mat...
 6.4.2: Let T1 : R2 R2 and T2 : R2 R2 be the linear transformations with ma...
 6.4.3: Let T1 : R2 R3 and T2 : R3 R2 be the linear transformations with ma...
 6.4.4: Let T1 : R2 R2 and T2 : R2 R2 be the linear transformations with ma...
 6.4.5: Let T1 : Mn(R) Mn(R) and T2 : Mn(R) Mn(R) be the linear transformat...
 6.4.6: Define T1 : C1[a, b] C0[a, b] and T2 : C0[a, b] C1[a, b] by T1( f )...
 6.4.7: Let {v1, v2} be a basis for the vector space V, and suppose that T1...
 6.4.8: Repeat under the assumption T1(v1) = 3v1 + v2, T1(v2) = 0 T2(v1) = ...
 6.4.9: Is the linear transformation T2T1 in Example 6.4.4 onetoone, onto,...
 6.4.10: For 1014, find Ker(T ) and Rng(T ), and hence, determine whether th...
 6.4.11: For 1014, find Ker(T ) and Rng(T ), and hence, determine whether th...
 6.4.12: For 1014, find Ker(T ) and Rng(T ), and hence, determine whether th...
 6.4.13: For 1014, find Ker(T ) and Rng(T ), and hence, determine whether th...
 6.4.14: For 1014, find Ker(T ) and Rng(T ), and hence, determine whether th...
 6.4.15: Suppose T : R3 R2 is a linear transformation such that T (1, 0, 0) ...
 6.4.16: Let V be a vector space and define T : V V by T (x) = x, where is a...
 6.4.17: Define T : P1(R) P1(R) by T (ax + b) = (2b a)x + (b + a). Show that...
 6.4.18: Define T : P2(R) P1(R) by T (ax2 + bx + c) = (a b)x + c. Determine ...
 6.4.19: Define T : P2(R) R2 by T (ax2 + bx + c) = (a 3b + 2c, b c). Determi...
 6.4.20: Let V denote the vector space of 2 2 symmetric matrices and define ...
 6.4.21: Define T : R3 M2(R) by T (a, b, c) = a + 3c a b c 2a + b 0 . Determ...
 6.4.22: Define T : M2(R) P3(R) by T a b c d = (a b + d) + (2a + b)x + cx2 +...
 6.4.23: Let {v1, v2} be a basis for the vector space V, and suppose that T ...
 6.4.24: Let v1 and v2 be a basis for the vector space V, and suppose that T...
 6.4.25: Determine an isomorphism between R2 and the vector space P1(R).
 6.4.26: Determine an isomorphism between R3 and the subspace of M2(R) consi...
 6.4.27: Determine an isomorphism between R and the subspace of M2(R) consis...
 6.4.28: Determine an isomorphism between R3 and the subspace of M2(R) consi...
 6.4.29: Let V denote the vector space of all 4 4 upper triangular matrices....
 6.4.30: Let V denote the subspace of P8(R) consisting of all polynomials wh...
 6.4.31: Let V denote the vector space of all 3 3 skewsymmetric matrices ove...
 6.4.32: For 3235, an invertible linear transformation Rn Rn is given. Find ...
 6.4.33: For 3235, an invertible linear transformation Rn Rn is given. Find ...
 6.4.34: For 3235, an invertible linear transformation Rn Rn is given. Find ...
 6.4.35: For 3235, an invertible linear transformation Rn Rn is given. Find ...
 6.4.36: Referring to 3233, compute the matrix of T2T1 and the matrix of T1T2.
 6.4.37: Referring to 3435, compute the matrix of T4T3 and the matrix of T3T4.
 6.4.38: Let T1 : V1 V2 and T2 : V2 V3 be linear transformations. (a) Prove ...
 6.4.39: Complete the proof of Theorem 6.4.2 by verifying Equation (6.4.2).
 6.4.40: Prove Proposition 6.4.14.
 6.4.41: If T : V W is an invertible linear transformation (that is, T 1 exi...
 6.4.42: Prove that if T : V V is a onetoone linear transformation, and V ...
 6.4.43: Prove that if T : V W is a onetoone linear transformation and {v1...
 6.4.44: Suppose T : V W is a linear transformation and {w1, w2,..., wm} spa...
 6.4.45: Prove that if T : V W is a linear transformation with dim[W] = n = ...
 6.4.46: Let T1 : V V and T2 : V V be linear transformations and suppose tha...
 6.4.47: Prove that if T : V V is a linear transformation such that T 2 = 0 ...
Solutions for Chapter 6.4: Additional Properties of Linear Transformations
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 6.4: Additional Properties of Linear Transformations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations, edition: 4. Differential Equations was written by and is associated to the ISBN: 9780321964670. Since 47 problems in chapter 6.4: Additional Properties of Linear Transformations have been answered, more than 19211 students have viewed full stepbystep solutions from this chapter. Chapter 6.4: Additional Properties of Linear Transformations includes 47 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).

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.