 47.1: Write the slopeintercept form of an equation for the line that pas...
 47.2: Write the slopeintercept form of an equation for the line that pas...
 47.3: Write the slopeintercept form of an equation for the line that pas...
 47.4: Write the slopeintercept form of an equation for the line that pas...
 47.5: GARDENS A garden is in the shape of a quadrilateral with vertices A...
 47.6: Write the slopeintercept form of an equation for the line that pas...
 47.7: Write the slopeintercept form of an equation for the line that pas...
 47.8: Write the slopeintercept form of an equation for the line that pas...
 47.9: Write the slopeintercept form for an equation of a line that is pe...
 47.10: Write the slopeintercept form of an equation for the line that pas...
 47.11: Write the slopeintercept form of an equation for the line that pas...
 47.12: Write the slopeintercept form of an equation for the line that pas...
 47.13: Write the slopeintercept form of an equation for the line that pas...
 47.14: Write the slopeintercept form of an equation for the line that pas...
 47.15: Write the slopeintercept form of an equation for the line that pas...
 47.16: GEOMETRY A parallelogram is a quadrilateral in which opposite sides...
 47.17: GEOMETRY The line with equation y = 3x  4 contains side AC of righ...
 47.18: Determine whether y = 6x + 4 and y = _1 6 x are perpendicular. Exp...
 47.19: MAPS On a map, Elmwood Drive passes through R(4, 11) and S(0, 9) ...
 47.20: Write the slopeintercept form of an equation for the line that pas...
 47.21: Write the slopeintercept form of an equation for the line that pas...
 47.22: Write the slopeintercept form of an equation for the line that pas...
 47.23: Write the slopeintercept form of an equation for the line that pas...
 47.24: Write the slopeintercept form of an equation for the line that pas...
 47.25: Write the slopeintercept form of an equation for the line that pas...
 47.26: Find an equation for the line that has a yintercept of 2 and is p...
 47.27: Write an equation of the line that is perpendicular to the line thr...
 47.28: Determine whether the graphs of each pair of equations are parallel...
 47.29: Determine whether the graphs of each pair of equations are parallel...
 47.30: Determine whether the graphs of each pair of equations are parallel...
 47.31: GEOMETRY Determine the relationship between the diagonals AC and BD...
 47.32: Write an equation of the line that is parallel to the graph of y = ...
 47.33: CHALLENGE The line that passes through the points (3a, 4) and (1, ...
 47.34: OPEN ENDED Draw two segments on the coordinate plane that appear to...
 47.35: Writing in Math Illustrate how you can determine y O x y x 1 3 2 wh...
 47.36: Which equation represents a line that is perpendicular to the graph...
 47.37: REVIEW If JKL is similar to JNM what is the length of side a? L K J...
 47.38: TECHNOLOGY Would a scatter plot showing the relationship between th...
 47.39: Write the pointslope form of an equation for the line that passes ...
 47.40: Write the pointslope form of an equation for the line that passes ...
 47.41: Write the pointslope form of an equation for the line that passes ...
 47.42: ARCHITECTURE An architect is building a scale model of a sports com...
 47.43: Solve _6c  t 7 = b for c.
Solutions for Chapter 47: Geometry: Parallel and Perpendicular Lines
Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1)  1st Edition
ISBN: 9780078738227
Solutions for Chapter 47: Geometry: Parallel and Perpendicular Lines
Get Full SolutionsSince 43 problems in chapter 47: Geometry: Parallel and Perpendicular Lines have been answered, more than 35305 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. Chapter 47: Geometry: Parallel and Perpendicular Lines includes 43 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column space C (A) =
space of all combinations of the columns of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.