 25.1: Vocabulary Write the definition of proof in your own words.
 25.2: MultiStep Solve each equation. Write a justification for each step...
 25.3: MultiStep Solve each equation. Write a justification for each step...
 25.4: MultiStep Solve each equation. Write a justification for each step...
 25.5: MultiStep Solve each equation. Write a justification for each step...
 25.6: MultiStep Solve each equation. Write a justification for each step...
 25.7: MultiStep Solve each equation. Write a justification for each step...
 25.8: Nutrition Amys favorite breakfast cereal has 102 Calories per servi...
 25.9: Movie Rentals The equation C = $5.75 + $0.89m relates the number of...
 25.10: Write a justification for each step.AB = BC 5y + 6 = 2y + 21 3y + 6...
 25.11: Write a justification for each step.PQ + QR = PR3n + 25 = 9n 525 =...
 25.12: Identify the property that justifies each statement.AB AB
 25.13: Identify the property that justifies each statement.m1 = m2, and m2...
 25.14: Identify the property that justifies each statement.x = y, so y = x.
 25.15: Identify the property that justifies each statement.ST YZ , and YZ ...
 25.16: MultiStep Solve each equation. Write a justification for each step...
 25.17: MultiStep Solve each equation. Write a justification for each step...
 25.18: MultiStep Solve each equation. Write a justification for each step...
 25.19: MultiStep Solve each equation. Write a justification for each step...
 25.20: MultiStep Solve each equation. Write a justification for each step...
 25.21: MultiStep Solve each equation. Write a justification for each step...
 25.22: Ecology The equation T = 0.03c + 0.05b relates the numbers of cans ...
 25.23: Write a justification for each step.mXYZ = m2 + m34n  6 = 58 + (2n...
 25.24: Write a justification for each step.mWYV = m1 + m2 5n = 3 (n  2) +...
 25.25: Identify the property that justifies each statement.KL PR , so PR K...
 25.26: Identify the property that justifies each statement.412 = 412
 25.27: Identify the property that justifies each statement.If a = b and b ...
 25.28: Identify the property that justifies each statement.figure A figure A
 25.29: Estimation Round the numbers in the equation 2 (3.1x  0.87) = 94.3...
 25.30: Use the indicated property to complete each statement.Reflexive Pro...
 25.31: Use the indicated property to complete each statement.Transitive Pr...
 25.32: Use the indicated property to complete each statement.Symmetric Pro...
 25.33: Recreation The north campground is midwaybetween the Northpoint Ove...
 25.34: Business A computer repair technician charges $35for each job plus ...
 25.35: Finance Morgan spent a total of $1,733.65 on her car last year. She...
 25.36: Critical Thinking Use the definition of segment congruence and the ...
 25.37: This problem will prepare you for the Concept Connection on page 12...
 25.38: Write About It Compare the conclusion of a deductive proof and a co...
 25.39: Which could NOT be used to justify the statement AB CD ? Definition...
 25.40: A club membership costs $35 plus $3 each time t the member uses the...
 25.41: Which statement is true by the Reflexive Property of Equality?x = 3...
 25.42: Gridded Response In the triangle, m1 + m2 + m3 = 180. If m3 = 2m1 a...
 25.43: In the gate, PA = QB, QB = RA, and PA = 18 in.Find PR, and justify ...
 25.44: Critical Thinking Explain why there isno Addition Property of Congr...
 25.45: Algebra Justify each step in the solutionof the inequality 7  3x >...
 25.46: The members of a high school band have saved $600 for a trip. They ...
 25.47: Use a compass and straightedge to construct each of the following. ...
 25.48: Use a compass and straightedge to construct each of the following. ...
 25.49: Identify whether each conclusion uses inductive or deductive reason...
 25.50: Identify whether each conclusion uses inductive or deductive reason...
Solutions for Chapter 25: Algebraic Proof
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 25: Algebraic Proof
Get Full SolutionsThis textbook survival guide was created for the textbook: Geometry, edition: 1. Geometry was written by and is associated to the ISBN: 9780030923456. Since 50 problems in chapter 25: Algebraic Proof have been answered, more than 42124 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 25: Algebraic Proof includes 50 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.