- 2-7.1: Vocabulary Apply the vocabulary from this lesson to answer each que...
- 2-7.2: Vocabulary Apply the vocabulary from this lesson to answer each que...
- 2-7.3: Use the given flowchart proof to writea two-column proof.Given: 1 2...
- 2-7.4: Use the given two-column proof to writea flowchart proof.Given: 2 a...
- 2-7.9: Use the given paragraph proof to write a two-column proof.Given: 1 ...
- 2-7.10: Use the given two-column proof to write a paragraph proof.Given: 1 ...
- 2-7.11: Find each measure and name the theorem that justifies your answer.AB
- 2-7.12: Find each measure and name the theorem that justifies your answer.m2
- 2-7.13: Find each measure and name the theorem that justifies your answer.m3
- 2-7.14: Algebra Find the value of each variable.14
- 2-7.15: Algebra Find the value of each variable.15
- 2-7.16: Algebra Find the value of each variable.16
- 2-7.17: /////ERROR ANALYSIS///// Below are two drawings for the given proof...
- 2-7.18: This problem will prepare you for the Concept Connection on page 12...
- 2-7.19: Critical Thinking Two lines intersect, and one of the angles formed...
- 2-7.20: Write About It Which style of proof do you find easiest to write? t...
- 2-7.21: Which pair of angles in the diagram must be congruent?1 and 5 5 and...
- 2-7.22: What is the measure of 2? 38 128 52 142
- 2-7.23: Which statement is NOT true if 2 and 6 are supplementary? m2 + m6 =...
- 2-7.24: Textiles Use the woven pattern to write a flowchart proof.Given: 1 ...
- 2-7.25: Write a two-column proof.Given: AOC BODProve: AOB COD
- 2-7.26: Write a paragraph proof.Given: 2 and 5 are right angles. m1 + m2 + ...
- 2-7.27: Multi-Step Find the value of each variable and the measures of all ...
- 2-7.28: Solve each system of equations. Check your solution. (Previous cour...
- 2-7.29: Solve each system of equations. Check your solution. (Previous cour...
- 2-7.30: Solve each system of equations. Check your solution. (Previous cour...
- 2-7.31: Use a protractor to draw an angle with each of the following measur...
- 2-7.32: Use a protractor to draw an angle with each of the following measur...
- 2-7.33: Use a protractor to draw an angle with each of the following measur...
- 2-7.34: Use a protractor to draw an angle with each of the following measur...
- 2-7.35: For each conditional, write the converse and a biconditional statem...
- 2-7.36: For each conditional, write the converse and a biconditional statem...
Solutions for Chapter 2-7: Flowchart and Paragraph Proofs
Full solutions for Geometry | 1st Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
A = CTC = (L.J]))(L.J]))T for positive definite A.
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.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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.
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!).
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.