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# Solutions for Chapter 7.7: Applications of Rational Expressions

## Full solutions for Beginning Algebra | 11th Edition

ISBN: 9780321673480

Solutions for Chapter 7.7: Applications of Rational Expressions

Solutions for Chapter 7.7
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##### ISBN: 9780321673480

This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. Since 52 problems in chapter 7.7: Applications of Rational Expressions have been answered, more than 39698 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Beginning Algebra was written by and is associated to the ISBN: 9780321673480. Chapter 7.7: Applications of Rational Expressions includes 52 full step-by-step solutions.

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!

• 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.

• Cayley-Hamilton Theorem.

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

• 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).

• Cramer's Rule for Ax = b.

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

• 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].

• 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

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• 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.

• Jordan form 1 = M- 1 AM.

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• 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 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.

• Projection matrix P onto subspace S.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

• Singular matrix A.

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

• Stiffness matrix

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

• 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.

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