×
×

# Solutions for Chapter 2: Reasoning in Geometry

## Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition

ISBN: 9781559538824

Solutions for Chapter 2: Reasoning in Geometry

Solutions for Chapter 2
4 5 0 300 Reviews
17
2
##### ISBN: 9781559538824

This expansive textbook survival guide covers the following chapters and their solutions. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Since 25 problems in chapter 2: Reasoning in Geometry have been answered, more than 25018 students have viewed full step-by-step solutions from this chapter. Chapter 2: Reasoning in Geometry includes 25 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Covariance matrix:E.

When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• 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.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• Left nullspace N (AT).

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

• Linear combination cv + d w or L C jV j.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• Plane (or hyperplane) in Rn.

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

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Row picture of Ax = b.

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

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

×