×

×

Textbooks / Math / Precalculus Enhanced with Graphing Utilities 6

# Precalculus Enhanced with Graphing Utilities 6th Edition - Solutions by Chapter ## Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition

ISBN: 9780132854351 Precalculus Enhanced with Graphing Utilities | 6th Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 305 Reviews
##### ISBN: 9780132854351

This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. The full step-by-step solution to problem in Precalculus Enhanced with Graphing Utilities were answered by , our top Math solution expert on 01/11/18, 01:38PM. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. Since problems from 114 chapters in Precalculus Enhanced with Graphing Utilities have been answered, more than 59560 students have viewed full step-by-step answer. This expansive textbook survival guide covers the following chapters: 114.

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!

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

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

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

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Dimension of vector space

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

• Hilbert matrix hilb(n).

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

• Identity matrix I (or In).

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

• 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 .

• Length II x II.

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

• Linear combination cv + d w or L C jV j.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• 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.

• Singular matrix A.

A square matrix that has no inverse: det(A) = o.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.