 Chapter 1: QUADRATIC EQUATIONS
 Chapter 10: PERIODIC BEHAVIOUR
 Chapter 11: TRIGONOMETRIC EQUATIONS
 Chapter 12: VECTORS AND SCALARS
 Chapter 13: PROBLEMS INVOLVING VECTOR OPERATIONS
 Chapter 14: LIMITS
 Chapter 15: Rules of differentiation
 Chapter 16: Properties of curves
 Chapter 17: Applications of differential calculus
 Chapter 18: Integration
 Chapter 19: Applications of integration
 Chapter 2: RELATIONS AND FUNCTIONS
 Chapter 20: Descriptive statistics
 Chapter 21: Linear modelling
 Chapter 22: Probability
 Chapter 23: Discrete random variables
 Chapter 24: The normal distribution
 Chapter 25: Miscellaneous questions
 Chapter 3: EXPONENTS
 Chapter 4: LOGARITHMS IN BASE 10
 Chapter 5: GRAPHING FUNCTIONS
 Chapter 6: NUMBER SEQUENCES
 Chapter 7: BINOMIAL EXPANSIONS
 Chapter 8: RADIAN MEASURE
 Chapter 9: AREAS OF TRIANGLES
Mathematics for the International Student: Mathematics SL 3rd Edition  Solutions by Chapter
Full solutions for Mathematics for the International Student: Mathematics SL  3rd Edition
ISBN: 9781921972089
Mathematics for the International Student: Mathematics SL  3rd Edition  Solutions by Chapter
Get Full SolutionsSince problems from 25 chapters in Mathematics for the International Student: Mathematics SL have been answered, more than 1897 students have viewed full stepbystep answer. This textbook survival guide was created for the textbook: Mathematics for the International Student: Mathematics SL, edition: 3. Mathematics for the International Student: Mathematics SL was written by and is associated to the ISBN: 9781921972089. The full stepbystep solution to problem in Mathematics for the International Student: Mathematics SL were answered by , our top Math solution expert on 03/15/18, 06:04PM. This expansive textbook survival guide covers the following chapters: 25.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= 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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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.
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