×

×

Textbooks / Math / College Algebra 9

# College Algebra 9th Edition - Solutions by Chapter

## Full solutions for College Algebra | 9th Edition

ISBN: 9780321716811

College Algebra | 9th Edition - Solutions by Chapter

Solutions by Chapter
4 5 0 324 Reviews
##### ISBN: 9780321716811

The full step-by-step solution to problem in College Algebra were answered by , our top Math solution expert on 03/19/18, 03:33PM. College Algebra was written by and is associated to the ISBN: 9780321716811. This expansive textbook survival guide covers the following chapters: 68. Since problems from 68 chapters in College Algebra have been answered, more than 60692 students have viewed full step-by-step answer. This textbook survival guide was created for the textbook: College Algebra, edition: 9.

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.

• 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.

• 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).

• 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].

• Determinant IAI = det(A).

Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

• Dimension of vector space

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

• Hermitian matrix A H = AT = A.

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

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Network.

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

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Particular solution x p.

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

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• Singular matrix A.

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

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• 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.

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

• Vector v in Rn.

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