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# Solutions for Chapter 10.4: The Parabola; Identifying Conic Sections

## Full solutions for Intermediate Algebra for College Students | 6th Edition

ISBN: 9780321758934

Solutions for Chapter 10.4: The Parabola; Identifying Conic Sections

Solutions for Chapter 10.4
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##### ISBN: 9780321758934

Chapter 10.4: The Parabola; Identifying Conic Sections includes 119 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 119 problems in chapter 10.4: The Parabola; Identifying Conic Sections have been answered, more than 9077 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934.

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.

• Back substitution.

Upper triangular systems are solved in reverse order Xn to Xl.

• 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).

• Column space C (A) =

space of all combinations of the columns of A.

• Complex conjugate

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

• 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.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• Left nullspace N (AT).

Nullspace of AT = "left nullspace" of A because y T A = OT.

• Length II x II.

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

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• Particular solution x p.

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

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Pseudoinverse A+ (Moore-Penrose inverse).

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Right inverse A+.

If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• 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!

• Singular matrix A.

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

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Toeplitz matrix.

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

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