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Differential Equations and Linear Algebra 3rd Edition - Solutions by Chapter

Full solutions for Differential Equations and Linear Algebra | 3rd Edition

ISBN: 9780136054252

Differential Equations and Linear Algebra | 3rd Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 405 Reviews
ISBN: 9780136054252

Differential Equations and Linear Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9780136054252. Since problems from 20 chapters in Differential Equations and Linear Algebra have been answered, more than 4287 students have viewed full step-by-step answer. This expansive textbook survival guide covers the following chapters: 20. The full step-by-step solution to problem in Differential Equations and Linear Algebra were answered by Sieva Kozinsky, our top Math solution expert on 08/31/17, 10:46AM. This textbook survival guide was created for the textbook: Differential Equations and Linear Algebra, edition: 3rd.

Key Math Terms and definitions covered in this textbook
• Augmented matrix [A b].

Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

• Cofactor Cij.

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

• Column space C (A) =

space of all combinations of the columns of A.

• 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.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Lucas numbers

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

• 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).

• 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.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Outer product uv T

= column times row = rank one matrix.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Transpose matrix AT.

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

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