 13.1.1: What is the difference between a postulate and a theorem?
 13.1.2: Euclid might have stated the addition property of equality (transla...
 13.1.3: Write the reflexive property of congruence, the transitive property...
 13.1.4: When you state AC = AC,what property are you using? When you state ...
 13.1.5: Name the property that supports this statement: If ACE BDF and BDF ...
 13.1.6: Name the property that supports this statement: If x + 120 = 180, t...
 13.1.7: Name the property that supports this statement: If 2(x + 14) = 36, ...
 13.1.8: In Exercises 8 and 9, provide the missing property of equality or a...
 13.1.9: In Exercises 8 and 9, provide the missing property of equality or a...
 13.1.10: In Exercises 1017, identify each statement as true or false. Then s...
 13.1.11: In Exercises 1017, identify each statement as true or false. Then s...
 13.1.12: In Exercises 1017, identify each statement as true or false. Then s...
 13.1.13: In Exercises 1017, identify each statement as true or false. Then s...
 13.1.14: In Exercises 1017, identify each statement as true or false. Then s...
 13.1.15: In Exercises 1017, identify each statement as true or false. Then s...
 13.1.16: In Exercises 1017, identify each statement as true or false. Then s...
 13.1.17: In Exercises 1017, identify each statement as true or false. Then s...
 13.1.18: The Declaration of Independence states, We hold these truths to be ...
 13.1.19: Copy and complete this flowchart proof. For each reason, state the ...
 13.1.20: For Exercises 2022, copy and complete each flowchart proof.
 13.1.21: For Exercises 2022, copy and complete each flowchart proof.
 13.1.22: For Exercises 2022, copy and complete each flowchart proof.
 13.1.23: You have probably noticed that the sum of two odd integers is alway...
 13.1.24: Let 2n 1 and 2m 1 represent any two odd integers, and prove that th...
 13.1.25: Show that the sum of any three consecutive integers is always divis...
 13.1.26: Shannon and Erin are hiking up a mountain. Of course, they are pack...
 13.1.27: Arrange the names of the solids in order, greatest to least. Volume...
 13.1.28: Two communication towers stand 64 ft apart. One is 80 ft high and t...
 13.1.29: In Exercises 29 and 30, all length measurements are given in meters...
 13.1.30: In Exercises 29 and 30, all length measurements are given in meters...
 13.1.31: Each arc is a quarter of a circle with its center at a vertex of th...
Solutions for Chapter 13.1: The Premises of Geometry
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 13.1: The Premises of Geometry
Get Full SolutionsSince 31 problems in chapter 13.1: The Premises of Geometry have been answered, more than 23382 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 13.1: The Premises of Geometry includes 31 full stepbystep solutions.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

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 SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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