 4.6.1: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.2: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.3: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.4: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.5: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.6: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.7: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.8: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.9: eveloping Proof For Exercises 19, copy the figures onto your paper ...
 4.6.10: For Exercises 10 and 11, you can use the right angles and the lengt...
 4.6.11: For Exercises 10 and 11, you can use the right angles and the lengt...
 4.6.12: In Chapter 3, you used inductive reasoning to discover how to dupli...
 4.6.13: In Exercises 1315, complete each statement. If the figure does not ...
 4.6.14: In Exercises 1315, complete each statement. If the figure does not ...
 4.6.15: In Exercises 1315, complete each statement. If the figure does not ...
 4.6.16: Construction Draw a triangle. Use the SAS Congruence Conjecture to ...
 4.6.17: Construction Construct two triangles that are not congruent, even t...
 4.6.18: Developing Proof Copy the figure. Calculate the measure of each let...
 4.6.19: According to math legend, the Greek mathematician Thales (ca. 62554...
 4.6.20: Isosceles right triangle ABC has vertices A( 8, 2), B( 5, 3), and C...
 4.6.21: The SSS Congruence Conjecture explains why triangles are rigid stru...
 4.6.22: Line is parallel to AB. If P moves to the right along , which of th...
 4.6.23: Find the lengths x and y Each angle is a right angle.
Solutions for Chapter 4.6: Corresponding Parts of Congruent Triangles
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 4.6: Corresponding Parts of Congruent Triangles
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Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

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

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.