×
×

# Solutions for Chapter 3.4: Polynomial and Rational Functions ## Full solutions for Algebra and Trigonometry: Real Mathematics, Real People | 7th Edition

ISBN: 9781305071735 Solutions for Chapter 3.4: Polynomial and Rational Functions

Solutions for Chapter 3.4
4 5 0 430 Reviews
14
5
##### ISBN: 9781305071735

Chapter 3.4: Polynomial and Rational Functions includes 78 full step-by-step solutions. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Since 78 problems in chapter 3.4: Polynomial and Rational Functions have been answered, more than 59379 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Tv = Av + Vo = linear transformation plus shift.

• Big formula for n by n determinants.

Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

• 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).

• 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.

• 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.

• Fourier matrix F.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• 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.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Normal equation AT Ax = ATb.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

• Right inverse A+.

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

• Semidefinite matrix A.

(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

• Singular matrix A.

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

• Singular Value Decomposition

(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Spectral radius = max of IAi I.

• Trace of A

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

T- 1 has rank 1 above and below diagonal.

v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

• Volume of box.

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

×
{"ft_lang_label":"__label__en","ft_lang_prob":0.7875103,"math_prob":0.99296117,"size":9873,"snap":"2020-34-2020-40","text_gpt3_token_len":2883,"char_repetition_ratio":0.30803525,"word_repetition_ratio":0.3787142,"special_character_ratio":0.349134,"punctuation_ratio":0.2632202,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99960726,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-01T15:43:00Z\",\"WARC-Record-ID\":\"<urn:uuid:d571c185-34d3-4e0e-8d6b-48e7ff6abd5f>\",\"Content-Length\":\"67026\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:133e0fc9-98b4-4b6a-8c3d-a164a9a03a32>\",\"WARC-Concurrent-To\":\"<urn:uuid:18b5a4ab-5a4a-48fb-a58c-65ab88c1b9d2>\",\"WARC-IP-Address\":\"54.189.254.180\",\"WARC-Target-URI\":\"https://studysoup.com/tsg/math/276/algebra-and-trigonometry-real-mathematics-real-people/chapter/12150/3-4\",\"WARC-Payload-Digest\":\"sha1:QJGCDAJCN66Y2R6PNEBDP5HB5WWQK5WS\",\"WARC-Block-Digest\":\"sha1:ZFOR5GVFH4457VPMFYV4HULETJ3ZZI56\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402131777.95_warc_CC-MAIN-20201001143636-20201001173636-00686.warc.gz\"}"}