×

×

Textbooks / Math / Geometry (Holt McDougal Larson Geometry) 1

# Geometry (Holt McDougal Larson Geometry) 1st Edition - Solutions by Chapter

## Full solutions for Geometry (Holt McDougal Larson Geometry) | 1st Edition

ISBN: 9780618595402

Geometry (Holt McDougal Larson Geometry) | 1st Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 259 Reviews
##### ISBN: 9780618595402

The full step-by-step solution to problem in Geometry (Holt McDougal Larson Geometry) were answered by , our top Math solution expert on 03/02/18, 04:34PM. This expansive textbook survival guide covers the following chapters: 12. Geometry (Holt McDougal Larson Geometry) was written by and is associated to the ISBN: 9780618595402. This textbook survival guide was created for the textbook: Geometry (Holt McDougal Larson Geometry), edition: 1. Since problems from 12 chapters in Geometry (Holt McDougal Larson Geometry) have been answered, more than 40296 students have viewed full step-by-step answer.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• 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).

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Diagonal matrix D.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• Graph G.

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.

• Hypercube matrix pl.

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

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Lucas numbers

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

• Matrix multiplication AB.

The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

• Multiplier eij.

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• 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.

• Subspace S of V.

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