- 3.1: A table is placed in the room. The legs of thetable are perpendicul...
- 3.2: Find the angle of the table top relative to theground. Suppose a ba...
- 3.3: Two people of the same height are standing onopposite ends of a boa...
- 3.4: In the room, a lamp hangs from the ceilingalong a line perpendicula...
- 3.5: Give an example of each angle pair.alternate interior angles
- 3.6: Give an example of each angle pair.alternate exterior angles
- 3.7: Give an example of each angle pair.corresponding angles
- 3.8: Give an example of each angle pair.same-side interior angles
- 3.9: Find each angle measure.1
- 3.10: Find each angle measure.2
- 3.11: Find each angle measure.3
- 3.12: Use the given information and the theorems and postulatesyou have l...
- 3.13: Use the given information and the theorems and postulatesyou have l...
- 3.14: Use the given information and the theorems and postulatesyou have l...
- 3.15: Use the given information and the theorems and postulatesyou have l...
- 3.16: Use the given information and the theorems and postulatesyou have l...
- 3.17: Write a two-column proof. Given: 1 2, nProve: p
- 3.18: Graph each line.y + 3 =_14 (x - 4)
- 3.19: Graph each line.x = 3
- 3.20: Write the equation of each line.7
- 3.21: Write the equation of each line.8
- 3.22: Write the equation of each line.9
- 3.23: Determine whether the lines are parallel, intersect, or coincide. y...
- 3.24: Determine whether the lines are parallel, intersect, or coincide. 3...
- 3.25: Determine whether the lines are parallel, intersect, or coincide. y...
Solutions for Chapter 3: Parallel and Perpendicular Lines
Full solutions for Geometry | 1st Edition
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
A sequence of steps intended to approach the desired solution.
lA-II = 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.
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.
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.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Solvable system Ax = b.
The right side b is in the column space of A.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).