- 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
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).
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 S-I 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.
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, ... , Aj-Ib. 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+ (Moore-Penrose 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 = M-I 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·