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Textbooks / Math / Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus 7

# Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus 7th Edition Solutions

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ISBN: 9781305253636

Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus | 7th Edition - Solutions by Chapter

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## Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus 7th Edition Student Assesment

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##### ISBN: 9781305253636

The full step-by-step solution to problem in Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus were answered by , our top Math solution expert on 10/03/18, 03:08PM. Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus was written by and is associated to the ISBN: 9781305253636. Since problems from 0 chapters in Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus have been answered, more than 200 students have viewed full step-by-step answer. This textbook survival guide was created for the textbook: Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus, edition: 7. This expansive textbook survival guide covers the following chapters: 0.

Key Math Terms and definitions covered in this textbook
• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• 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.

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Dimension of vector space

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

• 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.

• 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!).

• 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.

• Kirchhoff's Laws.

Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• Multiplicities AM and G M.

The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

• Particular solution x p.

Any solution to Ax = b; often x p has free variables = o.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Reduced row echelon form R = rref(A).

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.