 3.6.1: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.2: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.3: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.4: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.5: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.6: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.7: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.8: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.9: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.10: Construction In Exercises 110, first sketch and label the figure yo...
 3.6.11: Technology Using geometry software, draw a large scalene obtuse tri...
 3.6.12: Draw the new position of TEA if it is reflected over the dotted lin...
 3.6.13: Draw each figure and decide how many reflectional and rotational sy...
 3.6.14: Sketch the threedimensional figure formed by folding the net at ri...
 3.6.15: If a polygon has 500 diagonals from each vertex, how many sides doe...
 3.6.16: Use your geometry tools to draw parallelogram CARE so that CA 5.5 c...
Solutions for Chapter 3.6: Construction Problems
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 3.6: Construction Problems
Get Full SolutionsDiscovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Since 16 problems in chapter 3.6: Construction Problems have been answered, more than 8748 students have viewed full stepbystep solutions from this chapter. Chapter 3.6: Construction Problems includes 16 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.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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