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# Solutions for Chapter 1-1: Expressions and Formulas

## Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition

ISBN: 9780078738302

Solutions for Chapter 1-1: Expressions and Formulas

Solutions for Chapter 1-1
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##### ISBN: 9780078738302

Chapter 1-1: Expressions and Formulas includes 49 full step-by-step solutions. Since 49 problems in chapter 1-1: Expressions and Formulas have been answered, more than 53434 students have viewed full step-by-step solutions from this chapter. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1.

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.

• Cofactor Cij.

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

• Complex conjugate

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

• 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.

• Elimination.

A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

• 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!).

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• 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.

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

• 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).

• Pivot columns of A.

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

• Simplex method for linear programming.

The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

• Special solutions to As = O.

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

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• Subspace S of V.

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

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

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

T- 1 has rank 1 above and below diagonal.

• 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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