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# Solutions for Chapter 13.8: Coordinate Proof

## Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition

ISBN: 9781559538824

Solutions for Chapter 13.8: Coordinate Proof

Solutions for Chapter 13.8
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##### ISBN: 9781559538824

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. Chapter 13.8: Coordinate Proof includes 13 full step-by-step solutions. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. Since 13 problems in chapter 13.8: Coordinate Proof have been answered, more than 24947 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• 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.

• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• 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].

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

• Factorization

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.

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Gram-Schmidt 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.

• 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.

• 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 .

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• 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)ยท(b - Ax) = o.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• 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.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Transpose matrix AT.

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

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