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# 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

Solutions for Chapter 6.4
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##### ISBN: 9780321964670

This 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 step-by-step solutions from this chapter. Chapter 6.4: Additional Properties of Linear Transformations includes 47 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• 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. Block-diagonal: 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 = time-invariant (shift-invariant) 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.

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