- 2.1: My dad is in the navy, and he says that food is great on submarines...
- 2.2: Think of a situation you observed outside of school in which induct...
- 2.3: Think of a situation you observed outside of school in which deduct...
- 2.4: For Exercises 4 and 5, find the next two terms in the sequence. 7, ...
- 2.5: For Exercises 4 and 5, find the next two terms in the sequence. A, ...
- 2.6: For Exercises 6 and 7, generate the first six terms in the sequence...
- 2.7: For Exercises 6 and 7, generate the first six terms in the sequence...
- 2.8: For Exercises 8 and 9, draw the next shape in the pattern
- 2.9: For Exercises 8 and 9, draw the next shape in the pattern
- 2.10: For Exercises 10 and 11, look for a pattern and complete the conjec...
- 2.11: For Exercises 10 and 11, look for a pattern and complete the conjec...
- 2.12: For Exercises 12 and 13, find the nth term and the 20th term in the...
- 2.13: For Exercises 12 and 13, find the nth term and the 20th term in the...
- 2.14: Viktoriya is a store window designer for Savant Toys. She plans to ...
- 2.15: The stack of bricks at right is four bricks high. Find the total nu...
- 2.16: If at a party there are a total of 741 handshakes and each person s...
- 2.17: If a whole bunch of lines (no two parallel, no three concurrent) in...
- 2.18: If in a 54sided polygon all possible diagonals are drawn from one v...
- 2.19: In the diagram at right, a. Name a pair of vertical angles. b. Name...
- 2.20: Developing Proof In Exercise 19, name three angles congruent to and...
- 2.21: Consider this statement: If two polygons are congruent, then they h...
- 2.22: Using a ruler and a protractor, draw a parallelogram with one inter...
- 2.23: Developing Proof Which pairs of lines are parallel? Explain how you...
- 2.24: Developing Proof Whats wrong with this picture?
- 2.25: Developing Proof Trace the diagram at right. Calculate each lettere...
Solutions for Chapter 2: Reasoning in Geometry
Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
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.
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.
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.
Vector addition and scalar multiplication.
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.
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 ].
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).