- 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
peA) = det(A - AI) has peA) = zero 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 n-l - An).
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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.
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.
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 = Q-l. 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 = Q-1 = Q.
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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