- Chapter 1: A Game and Some Geometry
- Chapter 10: What Construction Means
- Chapter 11: Areas of Rectangles
- Chapter 12: Prisms
- Chapter 13: The Distance Formula
- Chapter 14: Mappings and Functions
- Chapter 2: If-Then Statements; Converses
- Chapter 3: Definitions
- Chapter 4: Congruent Figures
- Chapter 5: Properties of Parallelograms
- Chapter 6: Inequalities
- Chapter 7: Ratio and Proportion
- Chapter 8: Similarity in Right Triangles
- Chapter 9: Basic Terms
Geometry 1st Edition - Solutions by Chapter
Full solutions for Geometry | 1st Edition
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Column space C (A) =
space of all combinations of the columns of A.
A symmetric matrix with eigenvalues of both signs (+ and - ).
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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).
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