 4.1: Verify all the assertions in 4.5 except the last one
 4.2: Suppose m is a positive integer. Is the setf0g[fp 2 P.F/ W deg p D ...
 4.3: Is the setf0g[fp 2 P.F/ W deg p is evenga subspace of P.F/?
 4.4: Suppose m and n are positive integers with m n, and suppose1;:::;m ...
 4.5: Suppose m is a nonnegative integer, z1;:::;zmC1 are distinct elemen...
 4.6: Suppose p 2 P.C/ has degree m. Prove that p has m distinct zeros if...
 4.7: Prove that every polynomial of odd degree with real coefficients ha...
 4.8: Define T W P.R/ ! RR byTp D8<:p p.3/x 3if x 3;p0.3/ if x D 3:Show t...
 4.9: Suppose p 2 P.C/. Define q W C ! C byq.z/ D p.z/p.z/: NProve that q...
 4.10: Suppose m is a nonnegative integer and p 2 Pm.C/ is such that there...
 4.11: Suppose p 2 P.F/ with p 0. Let U D fpq W q 2 P.F/g.(a) Show that di...
Solutions for Chapter 4: Polynomials
Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics)  3rd Edition
ISBN: 9783319110790
Solutions for Chapter 4: Polynomials
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra Done Right (Undergraduate Texts in Mathematics), edition: 3. Chapter 4: Polynomials includes 11 full stepbystep solutions. Since 11 problems in chapter 4: Polynomials have been answered, more than 6676 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra Done Right (Undergraduate Texts in Mathematics) was written by and is associated to the ISBN: 9783319110790.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.