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Textbooks / Math / Elementary Number Theory 7

# Elementary Number Theory 7th Edition - Solutions by Chapter

## Full solutions for Elementary Number Theory | 7th Edition

ISBN: 9780073383149

Elementary Number Theory | 7th Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 264 Reviews
##### ISBN: 9780073383149

The full step-by-step solution to problem in Elementary Number Theory were answered by , our top Math solution expert on 03/14/18, 05:19PM. Elementary Number Theory was written by and is associated to the ISBN: 9780073383149. Since problems from 16 chapters in Elementary Number Theory have been answered, more than 14862 students have viewed full step-by-step answer. This textbook survival guide was created for the textbook: Elementary Number Theory, edition: 7. This expansive textbook survival guide covers the following chapters: 16.

Key Math Terms and definitions covered in this textbook
• Big formula for n by n determinants.

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Cholesky factorization

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

• 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).

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Diagonalizable matrix A.

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

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• 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 .

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• 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.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.