- 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
Mathematics for the International Student: Mathematics SL | 3rd Edition - Solutions by ChapterGet Full Solutions
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Column space C (A) =
space of all combinations of the columns of A.
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).
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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.
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
lA-II = 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.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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).
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.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).
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