- 0.2.1: What are the classical construction tools of geometry?
- 0.2.2: Create a line design from this lesson. Color your design.
- 0.2.3: Each of these line designs uses straight lines only. Select one des...
- 0.2.4: Describe the symmetries of the three designs in Exercise 3. For the...
- 0.2.5: Many quilt designers create beautiful geometric patterns with refle...
- 0.2.6: Geometric patterns seem to be in motion in a quilt design with rota...
- 0.2.7: Organic molecules have geometric shapes. How many different lines o...
Solutions for Chapter 0.2: Line Designs
Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
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.)
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A sequence of steps intended to approach the desired solution.
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.
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.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Constant down each diagonal = time-invariant (shift-invariant) filter.