 10.1.1: Questions 18 are about operations on complex numbers.
 10.1.2: Questions 18 are about operations on complex numbers.
 10.1.3: Questions 18 are about operations on complex numbers.
 10.1.4: Questions 18 are about operations on complex numbers.
 10.1.5: Questions 18 are about operations on complex numbers.
 10.1.6: Questions 18 are about operations on complex numbers.
 10.1.7: Questions 18 are about operations on complex numbers.
 10.1.8: Questions 18 are about operations on complex numbers.
 10.1.9: Questions 916 are about the conjugate z = a  ib = rei9 = z*.
 10.1.10: Questions 916 are about the conjugate z = a  ib = rei9 = z*.
 10.1.11: Questions 916 are about the conjugate z = a  ib = rei9 = z*.
 10.1.12: Questions 916 are about the conjugate z = a  ib = rei9 = z*.
 10.1.13: Questions 916 are about the conjugate z = a  ib = rei9 = z*.
 10.1.14: Questions 916 are about the conjugate z = a  ib = rei9 = z*.
 10.1.15: Questions 916 are about the conjugate z = a  ib = rei9 = z*.
 10.1.16: Questions 916 are about the conjugate z = a  ib = rei9 = z*.
 10.1.17: Questions 1724 are aboutthe form re iiJ of the complex number r co...
 10.1.18: Questions 1724 are aboutthe form re iiJ of the complex number r co...
 10.1.19: Questions 1724 are aboutthe form re iiJ of the complex number r co...
 10.1.20: Questions 1724 are aboutthe form re iiJ of the complex number r co...
 10.1.21: Questions 1724 are aboutthe form re iiJ of the complex number r co...
 10.1.22: Questions 1724 are aboutthe form re iiJ of the complex number r co...
 10.1.23: Questions 1724 are aboutthe form re iiJ of the complex number r co...
 10.1.24: Questions 1724 are aboutthe form re iiJ of the complex number r co...
Solutions for Chapter 10.1: Complex Numbers
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 10.1: Complex Numbers
Get Full SolutionsThis textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 24 problems in chapter 10.1: Complex Numbers have been answered, more than 11509 students have viewed full stepbystep solutions from this chapter. Chapter 10.1: Complex Numbers includes 24 full stepbystep solutions. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

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!).

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.