 5.5.1: Use your new conjectures in the following exercises. In Exercises 1...
 5.5.2: Use your new conjectures in the following exercises. In Exercises 1...
 5.5.3: Use your new conjectures in the following exercises. In Exercises 1...
 5.5.4: Use your new conjectures in the following exercises. In Exercises 1...
 5.5.5: Use your new conjectures in the following exercises. In Exercises 1...
 5.5.6: Use your new conjectures in the following exercises. In Exercises 1...
 5.5.7: Construction Given side LA, side AS, and L, construct parallelogram...
 5.5.8: Construction Given side DR and diagonals DO and PR, construct paral...
 5.5.9: In Exercises 9 and 10, copy the vector diagram and draw the resulta...
 5.5.10: In Exercises 9 and 10, copy the vector diagram and draw the resulta...
 5.5.11: Find the coordinates of point M in parallelogram PRAM.
 5.5.12: Draw a quadrilateral. Make a copy of it. Draw a diagonal in the fir...
 5.5.13: Developing Proof Copy and complete the flowchart to show how the Pa...
 5.5.14: Study the sewing box pictured here. Sketch the box as viewed from t...
 5.5.15: Find the measures of the lettered angles in this tiling of regular ...
 5.5.16: Trace the figure below. Calculate the measure of each lettered angle.
 5.5.17: Find x and y. Explain.
 5.5.18: What is the measure of each angle in the isosceles trapezoid face o...
 5.5.19: Developing Proof Is XYW WYZ ? Explain.
 5.5.20: Sketch the section formed when this pyramid is sliced by the plane.
 5.5.21: Developing Proof Construct two segments that bisect each other. Con...
 5.5.22: Developing Proof Construct two intersecting circles. Connect the tw...
Solutions for Chapter 5.5: Properties of Parallelograms
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 5.5: Properties of Parallelograms
Get Full SolutionsSince 22 problems in chapter 5.5: Properties of Parallelograms have been answered, more than 23778 students have viewed full stepbystep solutions from this chapter. Chapter 5.5: Properties of Parallelograms includes 22 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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