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
Solutions by ChapterTextbook: Mathematical Proofs: A Transition to Advanced MathematicsEdition: 3Author: Gary Chartrand, Albert D. Polimeni, Ping ZhangISBN: 9780321797094
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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