Mathematical proofs a transition to advanced mathematics solutions pdf

MATHEMATICALPROOFS:

A TRANSITIONTO

ADVANCEDMATHEMATICS

SECONDEDITION

Gary Chartrand

Western Michigan University

Albert D. Polimeni

SUNY, College at Fredonia

Ping Zhang

Western Michigan University

ONLINEINSTRUCTOR’S

SOLUTIONSMANUAL

Chartrand_390547_ISM_ttl.qxd 9/26/07 2:49 PM Page 1

Since problems from 16 chapters in Mathematical Proofs: A Transition to Advanced Mathematics have been answered, more than 25803 students have viewed full step-by-step answer. This expansive textbook survival guide covers the following chapters: 16. Mathematical Proofs: A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780321797094. This textbook survival guide was created for the textbook: Mathematical Proofs: A Transition to Advanced Mathematics, edition: 3. The full step-by-step solution to problem in Mathematical Proofs: A Transition to Advanced Mathematics were answered by , our top Math solution expert on 03/15/18, 05:53PM.

• Affine transformation

  Tv = Av + Vo = linear transformation plus shift.

• 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.

• Block matrix.

  A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Cayley-Hamilton Theorem.

  peA) = det(A - AI) has peA) = zero matrix.

• Complex conjugate

  z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• 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].

• Cyclic shift

  S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

• Diagonalization

  A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• 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.

• lA-II = 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.

• Left inverse A+.

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

• Left nullspace N (AT).

  Nullspace of AT = "left nullspace" of A because y T A = OT.

• Linearly dependent VI, ... , Vn.

  A combination other than all Ci = 0 gives L Ci Vi = O.

• 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.

• 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 .

• Multiplicities AM and G M.

  The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Network.

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

• Plane (or hyperplane) in Rn.

  Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Row picture of Ax = b.

  Each equation gives a plane in Rn; the planes intersect at x.

• Special solutions to As = O.

  One free variable is Si = 1, other free variables = o.

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