 13.2.1: Which postulate(s) does the VA Theorem rely on?
 13.2.2: Which postulate(s) does the Triangle Sum Theorem rely on?
 13.2.3: If you need a parallel line in a proof, which postulate allows you ...
 13.2.4: If you need a perpendicular line in a proof, which postulate allows...
 13.2.5: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.6: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.7: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.8: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.9: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.10: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.11: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.12: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.13: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.14: In Exercises 514, write a paragraph proof or a flowchart proof of t...
 13.2.15: Prove that the acute angles in a right triangle are complementary. ...
 13.2.16: Draw a family tree of the Converse of the Alternate Exterior Angles...
 13.2.17: Suppose the top of a pyramid with volume 1107 cm3 is sliced off and...
 13.2.18: Abraham is building a dog house for his terrier. His plan is shown ...
 13.2.19: A triangle has vertices A(7, 4), B(3, 2), and C(4, 1). Find the coo...
Solutions for Chapter 13.2: Planning a Geometry Proof
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 13.2: Planning a Geometry Proof
Get Full SolutionsDiscovering 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. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Since 19 problems in chapter 13.2: Planning a Geometry Proof have been answered, more than 25055 students have viewed full stepbystep solutions from this chapter. Chapter 13.2: Planning a Geometry Proof includes 19 full stepbystep solutions.

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.

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.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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 .

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Solvable system Ax = b.
The right side b is in the column space of A.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·