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# Solutions for Chapter 8.4: Mathematical Induction

## Full solutions for Precalculus With Limits A Graphing Approach | 5th Edition

ISBN: 9780618851522

Solutions for Chapter 8.4: Mathematical Induction

Solutions for Chapter 8.4
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##### ISBN: 9780618851522

Since 68 problems in chapter 8.4: Mathematical Induction have been answered, more than 47357 students have viewed full step-by-step solutions from this chapter. Chapter 8.4: Mathematical Induction includes 68 full step-by-step solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5.

Key Math Terms and definitions covered in this textbook
• Back substitution.

Upper triangular systems are solved in reverse order Xn to Xl.

• Basis for V.

Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Free variable Xi.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• 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.

• Length II x II.

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

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• Solvable system Ax = b.

The right side b is in the column space of A.

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Subspace S of V.

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

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• 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.

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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