- Chapter 1: Identify Points, Lines, and Planes
- Chapter 10: Use Properties of Tangents
- Chapter 11: Areas of Triangles and Parallelograms
- Chapter 12: Explore Solids
- Chapter 2: Use Inductive Reasoning
- Chapter 3: Identify Pairs of Lines and Angles
- Chapter 4: Apply Triangle Sum Properties
- Chapter 5: Midsegment Theorem and Coordinate Proof
- Chapter 6: Ratios, Proportions, and the Geometric Mean
- Chapter 7: Apply the Pythagorean Theorem
- Chapter 8: Find Angle Measures in Polygons
- Chapter 9: Translate Figures and Use Vectors
Geometry (Holt McDougal Larson Geometry) 1st Edition - Solutions by Chapter
Full solutions for Geometry (Holt McDougal Larson Geometry) | 1st Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column space C (A) =
space of all combinations of the columns of A.
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.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
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.
Outer product uv T
= column times row = rank one matrix.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Solvable system Ax = b.
The right side b is in the column space of A.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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