 11.6.1: Is a triangle with side lengths of 9 cm, 16 cm, and 25 cm a right t...
 11.6.2: Plot these points on graph paper. a. Draw the segment between the t...
 11.6.3: Find the distance between each pair of points. a. (0, 0) and (3, 4)...
 11.6.4: On his homework, Matt wrote that the distance between two points wa...
 11.6.5: Quadrilateral ABCD is pictured at right.a. What is the slope of eac...
 11.6.6: For Exercises 6 and 7, refer to the Amusement Park investigation on...
 11.6.7: For Exercises 6 and 7, refer to the Amusement Park investigation on...
 11.6.8: APPLICATION The longest pole that fits in a rectangular box goes fr...
 11.6.9: MiniInvestigation Consider the equation a. Solve the equation symb...
 11.6.10: Solve each equation. a. b. c.
 11.6.11: APPLICATION Nadia Ferrell wants to build an awning over her porch. ...
 11.6.12: Rewrite each radical expression so that it contains no perfectsqua...
Solutions for Chapter 11.6: A Distance Formula
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 11.6: A Distance Formula
Get Full SolutionsSince 12 problems in chapter 11.6: A Distance Formula have been answered, more than 8426 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Chapter 11.6: A Distance Formula includes 12 full stepbystep solutions.

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).

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.