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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

Since problems from 25 chapters in Mathematics for the International Student: Mathematics SL have been answered, more than 3103 students have viewed full step-by-step 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 step-by-step 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.

Key Math Terms and definitions covered in this textbook
  • 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.

  • Distributive Law

    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

  • Fundamental Theorem.

    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.

  • Gauss-Jordan method.

    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).

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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