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Solutions for Chapter 2.2: Properties of Determinants

Full solutions for Linear Algebra with Applications | 8th Edition

ISBN: 9780136009290

Solutions for Chapter 2.2: Properties of Determinants

Solutions for Chapter 2.2
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Textbook: Linear Algebra with Applications
Edition: 8
Author: Steve Leon
ISBN: 9780136009290

This expansive textbook survival guide covers the following chapters and their solutions. Since 21 problems in chapter 2.2: Properties of Determinants have been answered, more than 17841 students have viewed full step-by-step solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. Chapter 2.2: Properties of Determinants includes 21 full step-by-step solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8.

Key Math Terms and definitions covered in this textbook
  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • 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!

  • Big formula for n by n determinants.

    Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Cramer's Rule for Ax = b.

    B j has b replacing column j of A; x j = det B j I det A

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • 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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.

  • Gauss-Jordan method.

    Invert A by row operations on [A I] to reach [I A-I].

  • Indefinite matrix.

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

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Orthogonal subspaces.

    Every v in V is orthogonal to every w in W.

  • Pascal matrix

    Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Row picture of Ax = b.

    Each equation gives a plane in Rn; the planes intersect at x.

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • Vandermonde matrix V.

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

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.