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# Solutions for Chapter 6.4: Special Factoring Techniques

## Full solutions for Beginning Algebra | 11th Edition

ISBN: 9780321673480

Solutions for Chapter 6.4: Special Factoring Techniques

Solutions for Chapter 6.4
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##### ISBN: 9780321673480

Since 102 problems in chapter 6.4: Special Factoring Techniques have been answered, more than 36376 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Beginning Algebra, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.4: Special Factoring Techniques includes 102 full step-by-step solutions. Beginning Algebra was written by and is associated to the ISBN: 9780321673480.

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.

• Cayley-Hamilton Theorem.

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

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

• Cofactor Cij.

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

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

• 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!).

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• 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.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• 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.

• Reduced row echelon form R = rref(A).

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Right inverse A+.

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

• Row picture of Ax = b.

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

• Subspace S of V.

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

• Sum V + W of subs paces.

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

• Vector v in Rn.

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

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