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# Solutions for Chapter 7.1: Algebra and Trigonometry 9th Edition

## Full solutions for Algebra and Trigonometry | 9th Edition

ISBN: 9780321716569

Solutions for Chapter 7.1

Solutions for Chapter 7.1
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##### ISBN: 9780321716569

This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. Since 125 problems in chapter 7.1 have been answered, more than 61429 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9. Chapter 7.1 includes 125 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

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

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

• Distributive Law

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

• 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).

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• 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.

• Hankel matrix H.

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

• 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).

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Outer product uv T

= column times row = rank one matrix.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

P = aaT laTa has rank l.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• 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.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• 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.

• Vector v in Rn.

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

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.

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