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# Solutions for Chapter 3.3: Systems of Linear Equations in Three Variables

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

ISBN: 9780321758934

Solutions for Chapter 3.3: Systems of Linear Equations in Three Variables

Solutions for Chapter 3.3
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##### ISBN: 9780321758934

Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. This expansive textbook survival guide covers the following chapters and their solutions. Since 71 problems in chapter 3.3: Systems of Linear Equations in Three Variables have been answered, more than 6772 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. Chapter 3.3: Systems of Linear Equations in Three Variables includes 71 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Complex conjugate

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

• Dimension of vector space

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

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• 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

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• 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.

• 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).

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• 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).

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• 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.

• Plane (or hyperplane) in Rn.

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

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• 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!

• Sum V + W of subs paces.

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

• 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.

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